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bg-background","children":["$","$L3b",null,{"backHref":"/a-level/maths/ocr/questions","basePath":"/a-level/maths/ocr/integration-by-substitution/questions","formulaUrl":null,"initialIndex":17,"initialQuestionWindow":[{"id":"1c062306-1388-4be0-af8a-f4aee2770406","revisionId":"33729c02-7b10-4ee5-b7ae-6a9961ed3d28","questionNumber":"2","questionType":"LA","specification":"Integrate with respect to $x$:","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":6,"sourceTitle":"Integration","calculatorUse":null,"isCalculation":true,"parts":[{"id":"0d1dbc9c-618f-4220-9924-d83058c1f13e","label":"a","content":"$$\\sqrt{5x - 3}$","markscheme":"- Rewrites as $(5x-3)^{\\frac{1}{2}}$ and integrates: $\\frac{2}{3}(5x-3)^{\\frac{3}{2}} \\times \\frac{1}{5} + C$ M1\n- Simplifies to $\\frac{2}{15}(5x-3)^{\\frac{3}{2}} + C$ A1\n- Includes constant of integration $+ C$ B1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Integrates to form $k(5x-3)^{\\frac{3}{2}}$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly applies the factor of $\\frac{1}{5}$.","code":"M"},{"id":"p3","marks":1,"description":"Final answer $\\frac{2}{15}(5x-3)^{\\frac{3}{2}} + C$.","code":"A"}],"finalAnswers":["\\frac{2}{15}(5x-3)^{\\frac{3}{2}} + C"],"generatedAt":"2026-05-06T03:37:59.137Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"reverse-chain-rule","label":"Reverse Chain Rule","steps":[{"type":"text","order":0,"title":"Rewrite the integral","content":"To integrate $\\sqrt{5x - 3}$ with respect to $x$, we first write it in index form. Recall that a square root can be written as a fractional exponent of $\\frac{1}{2}$:\n\n$$\\int \\sqrt{5x - 3} \\, dx = \\int (5x - 3)^{\\frac{1}{2}} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\int (\\htmlData{anchor=linear}{\\color{@sky}{5x - 3}})^{\\htmlData{anchor=power-old}{\\color{@amber}{\\frac{1}{2}}}} \\, dx$$"},{"latex":"$$$= \\frac{(\\color{@sky}{5x - 3})^{\\htmlData{anchor=power-new}{\\color{@amber}{\\frac{3}{2}}}}}{\\color{@amber}{\\frac{3}{2}}} \\times \\frac{1}{\\htmlData{anchor=coeff}{\\color{@purple}{5}}} + C$$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"power-old"},{"color":"amber","style":"text-color","anchor":"power-new"},{"color":"purple","style":"text-color","anchor":"coeff"}],"connections":[{"to":"power-new","from":"power-old","color":"amber","label":"Add 1 to power","style":"solid"},{"to":"coeff","from":"linear","color":"purple","label":"Divide by coefficient","style":"solid"}]},"order":1,"title":"Apply reverse chain rule","description":"We apply the reverse chain rule for linear compositions $\\int (ax + b)^n \\, dx = \\frac{(ax + b)^{n+1}}{a(n+1)} + C$. Here, we increase the power by $1$ to get $\\frac{3}{2}$, divide by this new power, and also divide by the coefficient of $x$, which is $5$."},{"type":"text","order":2,"title":"Simplify and add constant","content":"Let's simplify the constant coefficient of our integrated expression. \n\nDividing by $\\frac{3}{2}$ is equivalent to multiplying by $\\frac{2}{3}$. Let's combine this with the division by $5$:\n\n$$\\frac{2}{3} \\times \\frac{1}{5} = \\frac{2}{15}$$\n\nPutting it all together, we get our final integrated expression, remembering to include the constant of integration, $+ C$:\n\n$$\\frac{2}{15}(5x - 3)^{\\frac{3}{2}} + C$$"},{"type":"text","order":3,"title":"Markscheme breakdown","content":"Here is how the **3 marks** are awarded for this question:\n\n* **M1**: For rewriting the integrand as $(5x-3)^{\\frac{1}{2}}$ and attempting to integrate to get $\\frac{2}{3}(5x-3)^{\\frac{3}{2}} \\times \\frac{1}{5}$.\n* **A1**: For correctly simplifying the expression to $\\frac{2}{15}(5x-3)^{\\frac{3}{2}}$.\n* **B1**: For including the constant of integration, $+ C$."}]},{"id":"substitution","label":"Substitution Method","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We want to find the indefinite integral of $\\sqrt{5x - 3}$ with respect to $x$. First, let's write it down using the integration symbol and express the square root as a fractional power:\n\n$$\\int \\sqrt{5x - 3} \\, dx = \\int (5x - 3)^{\\frac{1}{2}} \\, dx$$","highlights":[{"text":"$$\\sqrt{5x - 3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"u = \\htmlData{anchor=sub-expr}{\\color{@sky}{5x - 3}}"},{"text":"Differentiating both sides with respect to $x$ gives:"},{"latex":"\\frac{du}{dx} = 5"},{"text":"Which we can rewrite to find an expression for $dx$:"},{"latex":"dx = \\htmlData{anchor=dx-expr}{\\color{@amber}{\\frac{1}{5} \\, du}}"}],"highlights":[{"color":"sky","style":"box","anchor":"sub-expr"},{"color":"amber","style":"box","anchor":"dx-expr"}],"connections":[]},"order":1,"title":"Choose the substitution","description":"We choose $u = 5x - 3$ to simplify the expression inside the square root."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\int (\\htmlData{anchor=u-term}{\\color{@sky}{5x - 3}})^{\\frac{1}{2}} \\, \\htmlData{anchor=dx-term}{\\color{@amber}{dx}}"},{"text":"Substituting our terms for $u$ and $dx$ yields:"},{"latex":"\\int \\htmlData{anchor=u-sub}{\\color{@sky}{u}}^{\\frac{1}{2}} \\cdot \\htmlData{anchor=dx-sub}{\\color{@amber}{\\frac{1}{5} \\, du}} = \\frac{1}{5} \\int u^{\\frac{1}{2}} \\, du"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-term"},{"color":"amber","style":"text-color","anchor":"dx-term"}],"connections":[{"to":"u-sub","from":"u-term","color":"sky","label":"substitute u","style":"solid"},{"to":"dx-sub","from":"dx-term","color":"amber","label":"substitute dx","style":"solid"}]},"order":2,"title":"Transform the integral","description":"Now, substitute $u$ and $dx$ into the original integral to write it entirely in terms of $u$."},{"type":"text","order":3,"title":"Integrate with respect to u","content":"Now we integrate the power of $u$ using the standard power rule for integration, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1} + C$:\n\n$$\\frac{1}{5} \\int u^{\\frac{1}{2}} \\, du = \\frac{1}{5} \\left( \\frac{u^{\\frac{3}{2}}}{\\frac{3}{2}} \\right) + C$$\n\nSimplifying the fraction:\n\n$$\\frac{1}{5} \\left( \\frac{2}{3} u^{\\frac{3}{2}} \\right) + C = \\frac{2}{15} u^{\\frac{3}{2}} + C$$\n\n*(Note: Remembering to include the constant of integration $+ C$ is required to earn full marks.)*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{2}{15} \\htmlData{anchor=u-back}{\\color{@sky}{u}}^{\\frac{3}{2}} + C"},{"text":"Replacing $u$ with our original expression $5x - 3$ gives:"},{"latex":"\\frac{2}{15} (\\htmlData{anchor=x-back}{\\color{@sky}{5x - 3}})^{\\frac{3}{2}} + C"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-back"}],"connections":[{"to":"x-back","from":"u-back","color":"sky","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to x","description":"Substitute $u = 5x - 3$ back into the integrated expression to get the final answer in terms of $x$."}]}],"generatedAt":"2026-06-09T03:32:19.011Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"89f7c1cd-8cab-4a4b-852d-5f1bb4ab1080","label":"b","content":"$$e^{2x+3}$","markscheme":"- Integrates exponential: $\\frac{1}{2}e^{2x+3} + C$ M1\n- Correct coefficient $\\frac{1}{2}$ A1\n- Includes constant of integration $+ C$ B1","marks":3,"order":1,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Recognises integral of $e^{ax+b}$ is $\\frac{1}{a}e^{ax+b}$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly identifies $a=2$.","code":"A"},{"id":"p3","marks":1,"description":"Final answer $\\frac{1}{2}e^{2x+3} + C$.","code":"B"}],"finalAnswers":["\\frac{1}{2}e^{2x+3} + C"],"generatedAt":"2026-05-06T03:37:59.137Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"reverse-chain-rule","label":"Reverse Chain Rule","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We need to find the indefinite integral of the exponential function:\n\n$$\\int e^{2x+3} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int e^{\\htmlData{anchor=linear}{\\color{@sky}{2}}x + 3} \\, dx$"},{"latex":"$$= \\frac{1}{\\htmlData{anchor=deriv}{\\color{@sky}{2}}} e^{\\htmlData{anchor=power}{\\color{@sky}{2}x + 3}} + \\htmlData{anchor=c-val}{\\color{@purple}{C}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"linear"},{"color":"sky","style":"text-color","anchor":"deriv"},{"color":"purple","style":"text-color","anchor":"c-val"}],"connections":[{"to":"deriv","from":"linear","color":"sky","label":"divide by derivative","style":"solid"},{"to":"power","from":"linear","color":"sky","label":"power remains","style":"dashed"}]},"order":1,"title":"Apply reverse chain rule","description":"For any exponential function with a linear exponent of the form $ax+b$, we divide the function by the derivative of the exponent, which is the constant $a$."},{"type":"text","order":2,"title":"Summarize marks","content":"Our final answer is:\n\n$$\\frac{1}{2}e^{2x+3} + C$$\n\n### Marks Breakdown\n- **M1**: For integrating the exponential term to get a form of $k e^{2x+3}$.\n- **A1**: For obtaining the correct coefficient of $\\frac{1}{2}$.\n- **B1**: For including the constant of integration, $+ C$."}]},{"id":"substitution","label":"Integration by Substitution","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We need to integrate the function $e^{2x+3}$ with respect to $x$. Let's write this as an indefinite integral:\n\n$$\\int e^{2x+3} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Let's choose our substitution variable $u$ to represent the linear expression in the exponent:"},{"latex":"$$\\htmlData{anchor=u-def}{\\color{@sky}{u}} = \\htmlData{anchor=exponent}{\\color{@sky}{2x + 3}}$"},{"text":"Now, we differentiate $u$ with respect to $x$ to find the differential term:"},{"latex":"$$\\frac{d\\htmlData{anchor=u-diff}{\\color{@sky}{u}}}{dx} = \\htmlData{anchor=deriv}{\\color{@amber}{2}}$"},{"text":"Rearranging this expression gives us the relationship for $dx$:"},{"latex":"$$\\htmlData{anchor=dx-val}{\\color{@purple}{dx}} = \\frac{1}{\\htmlData{anchor=deriv-sub}{\\color{@amber}{2}}} \\, \\htmlData{anchor=du-val}{\\color{@purple}{du}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-def"},{"color":"sky","style":"text-color","anchor":"exponent"},{"color":"amber","style":"text-color","anchor":"deriv"},{"color":"purple","style":"text-color","anchor":"dx-val"},{"color":"purple","style":"text-color","anchor":"du-val"}],"connections":[{"to":"deriv-sub","from":"deriv","color":"amber","label":"rearrange","style":"solid"}]},"order":1,"title":"Define the substitution","description":"We define a new variable $u$ to simplify the exponent of the exponential function."},{"type":"text","order":2,"title":"Substitute into the integral","content":"Now we substitute both $u = 2x + 3$ and $dx = \\frac{1}{2} \\, du$ back into the original integral:\n\n$$\\int e^{2x+3} \\, dx = \\int e^{u} \\left( \\frac{1}{2} \\, du \\right)$$\n\nBy pulling the constant factor $\\frac{1}{2}$ out of the integration sign, we get:\n\n$$\\frac{1}{2} \\int e^{u} \\, du$$"},{"type":"text","order":3,"title":"Integrate with respect to u","content":"The integral of $e^u$ is a standard integral from our formula booklet:\n\n$$\\int e^u \\, du = e^u + C$$\n\nApplying this standard form, we get:\n\n$$\\frac{1}{2} e^u + C$$"},{"type":"text","order":4,"title":"Substitute back to x","content":"Finally, we substitute $u = 2x+3$ back into our result to express our final answer in terms of $x$:\n\n$$\\frac{1}{2} e^{2x+3} + C$$\n\n### Markscheme Breakdown\n- **M1**: Integrating the exponential function to get the correct general form.\n- **A1**: Finding the correct coefficient $\\frac{1}{2}$.\n- **B1**: Including the constant of integration $+ C$."}]}],"generatedAt":"2026-06-09T03:33:44.353Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"0d1dbc9c-618f-4220-9924-d83058c1f13e:reverse-chain-rule","label":"(a) Reverse Chain Rule","steps":[{"type":"text","order":0,"title":"Rewrite the integral","content":"To integrate $\\sqrt{5x - 3}$ with respect to $x$, we first write it in index form. Recall that a square root can be written as a fractional exponent of $\\frac{1}{2}$:\n\n$$\\int \\sqrt{5x - 3} \\, dx = \\int (5x - 3)^{\\frac{1}{2}} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\int (\\htmlData{anchor=linear}{\\color{@sky}{5x - 3}})^{\\htmlData{anchor=power-old}{\\color{@amber}{\\frac{1}{2}}}} \\, dx$$"},{"latex":"$$$= \\frac{(\\color{@sky}{5x - 3})^{\\htmlData{anchor=power-new}{\\color{@amber}{\\frac{3}{2}}}}}{\\color{@amber}{\\frac{3}{2}}} \\times \\frac{1}{\\htmlData{anchor=coeff}{\\color{@purple}{5}}} + C$$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"power-old"},{"color":"amber","style":"text-color","anchor":"power-new"},{"color":"purple","style":"text-color","anchor":"coeff"}],"connections":[{"to":"power-new","from":"power-old","color":"amber","label":"Add 1 to power","style":"solid"},{"to":"coeff","from":"linear","color":"purple","label":"Divide by coefficient","style":"solid"}]},"order":1,"title":"Apply reverse chain rule","description":"We apply the reverse chain rule for linear compositions $\\int (ax + b)^n \\, dx = \\frac{(ax + b)^{n+1}}{a(n+1)} + C$. Here, we increase the power by $1$ to get $\\frac{3}{2}$, divide by this new power, and also divide by the coefficient of $x$, which is $5$."},{"type":"text","order":2,"title":"Simplify and add constant","content":"Let's simplify the constant coefficient of our integrated expression. \n\nDividing by $\\frac{3}{2}$ is equivalent to multiplying by $\\frac{2}{3}$. Let's combine this with the division by $5$:\n\n$$\\frac{2}{3} \\times \\frac{1}{5} = \\frac{2}{15}$$\n\nPutting it all together, we get our final integrated expression, remembering to include the constant of integration, $+ C$:\n\n$$\\frac{2}{15}(5x - 3)^{\\frac{3}{2}} + C$$"},{"type":"text","order":3,"title":"Markscheme breakdown","content":"Here is how the **3 marks** are awarded for this question:\n\n* **M1**: For rewriting the integrand as $(5x-3)^{\\frac{1}{2}}$ and attempting to integrate to get $\\frac{2}{3}(5x-3)^{\\frac{3}{2}} \\times \\frac{1}{5}$.\n* **A1**: For correctly simplifying the expression to $\\frac{2}{15}(5x-3)^{\\frac{3}{2}}$.\n* **B1**: For including the constant of integration, $+ C$."}]},{"id":"0d1dbc9c-618f-4220-9924-d83058c1f13e:substitution","label":"(a) Substitution Method","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We want to find the indefinite integral of $\\sqrt{5x - 3}$ with respect to $x$. First, let's write it down using the integration symbol and express the square root as a fractional power:\n\n$$\\int \\sqrt{5x - 3} \\, dx = \\int (5x - 3)^{\\frac{1}{2}} \\, dx$$","highlights":[{"text":"$$\\sqrt{5x - 3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"u = \\htmlData{anchor=sub-expr}{\\color{@sky}{5x - 3}}"},{"text":"Differentiating both sides with respect to $x$ gives:"},{"latex":"\\frac{du}{dx} = 5"},{"text":"Which we can rewrite to find an expression for $dx$:"},{"latex":"dx = \\htmlData{anchor=dx-expr}{\\color{@amber}{\\frac{1}{5} \\, du}}"}],"highlights":[{"color":"sky","style":"box","anchor":"sub-expr"},{"color":"amber","style":"box","anchor":"dx-expr"}],"connections":[]},"order":1,"title":"Choose the substitution","description":"We choose $u = 5x - 3$ to simplify the expression inside the square root."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\int (\\htmlData{anchor=u-term}{\\color{@sky}{5x - 3}})^{\\frac{1}{2}} \\, \\htmlData{anchor=dx-term}{\\color{@amber}{dx}}"},{"text":"Substituting our terms for $u$ and $dx$ yields:"},{"latex":"\\int \\htmlData{anchor=u-sub}{\\color{@sky}{u}}^{\\frac{1}{2}} \\cdot \\htmlData{anchor=dx-sub}{\\color{@amber}{\\frac{1}{5} \\, du}} = \\frac{1}{5} \\int u^{\\frac{1}{2}} \\, du"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-term"},{"color":"amber","style":"text-color","anchor":"dx-term"}],"connections":[{"to":"u-sub","from":"u-term","color":"sky","label":"substitute u","style":"solid"},{"to":"dx-sub","from":"dx-term","color":"amber","label":"substitute dx","style":"solid"}]},"order":2,"title":"Transform the integral","description":"Now, substitute $u$ and $dx$ into the original integral to write it entirely in terms of $u$."},{"type":"text","order":3,"title":"Integrate with respect to u","content":"Now we integrate the power of $u$ using the standard power rule for integration, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1} + C$:\n\n$$\\frac{1}{5} \\int u^{\\frac{1}{2}} \\, du = \\frac{1}{5} \\left( \\frac{u^{\\frac{3}{2}}}{\\frac{3}{2}} \\right) + C$$\n\nSimplifying the fraction:\n\n$$\\frac{1}{5} \\left( \\frac{2}{3} u^{\\frac{3}{2}} \\right) + C = \\frac{2}{15} u^{\\frac{3}{2}} + C$$\n\n*(Note: Remembering to include the constant of integration $+ C$ is required to earn full marks.)*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{2}{15} \\htmlData{anchor=u-back}{\\color{@sky}{u}}^{\\frac{3}{2}} + C"},{"text":"Replacing $u$ with our original expression $5x - 3$ gives:"},{"latex":"\\frac{2}{15} (\\htmlData{anchor=x-back}{\\color{@sky}{5x - 3}})^{\\frac{3}{2}} + C"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-back"}],"connections":[{"to":"x-back","from":"u-back","color":"sky","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to x","description":"Substitute $u = 5x - 3$ back into the integrated expression to get the final answer in terms of $x$."}]},{"id":"89f7c1cd-8cab-4a4b-852d-5f1bb4ab1080:reverse-chain-rule","label":"(b) Reverse Chain Rule","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We need to find the indefinite integral of the exponential function:\n\n$$\\int e^{2x+3} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int e^{\\htmlData{anchor=linear}{\\color{@sky}{2}}x + 3} \\, dx$"},{"latex":"$$= \\frac{1}{\\htmlData{anchor=deriv}{\\color{@sky}{2}}} e^{\\htmlData{anchor=power}{\\color{@sky}{2}x + 3}} + \\htmlData{anchor=c-val}{\\color{@purple}{C}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"linear"},{"color":"sky","style":"text-color","anchor":"deriv"},{"color":"purple","style":"text-color","anchor":"c-val"}],"connections":[{"to":"deriv","from":"linear","color":"sky","label":"divide by derivative","style":"solid"},{"to":"power","from":"linear","color":"sky","label":"power remains","style":"dashed"}]},"order":1,"title":"Apply reverse chain rule","description":"For any exponential function with a linear exponent of the form $ax+b$, we divide the function by the derivative of the exponent, which is the constant $a$."},{"type":"text","order":2,"title":"Summarize marks","content":"Our final answer is:\n\n$$\\frac{1}{2}e^{2x+3} + C$$\n\n### Marks Breakdown\n- **M1**: For integrating the exponential term to get a form of $k e^{2x+3}$.\n- **A1**: For obtaining the correct coefficient of $\\frac{1}{2}$.\n- **B1**: For including the constant of integration, $+ C$."}]},{"id":"89f7c1cd-8cab-4a4b-852d-5f1bb4ab1080:substitution","label":"(b) Integration by Substitution","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We need to integrate the function $e^{2x+3}$ with respect to $x$. Let's write this as an indefinite integral:\n\n$$\\int e^{2x+3} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Let's choose our substitution variable $u$ to represent the linear expression in the exponent:"},{"latex":"$$\\htmlData{anchor=u-def}{\\color{@sky}{u}} = \\htmlData{anchor=exponent}{\\color{@sky}{2x + 3}}$"},{"text":"Now, we differentiate $u$ with respect to $x$ to find the differential term:"},{"latex":"$$\\frac{d\\htmlData{anchor=u-diff}{\\color{@sky}{u}}}{dx} = \\htmlData{anchor=deriv}{\\color{@amber}{2}}$"},{"text":"Rearranging this expression gives us the relationship for $dx$:"},{"latex":"$$\\htmlData{anchor=dx-val}{\\color{@purple}{dx}} = \\frac{1}{\\htmlData{anchor=deriv-sub}{\\color{@amber}{2}}} \\, \\htmlData{anchor=du-val}{\\color{@purple}{du}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-def"},{"color":"sky","style":"text-color","anchor":"exponent"},{"color":"amber","style":"text-color","anchor":"deriv"},{"color":"purple","style":"text-color","anchor":"dx-val"},{"color":"purple","style":"text-color","anchor":"du-val"}],"connections":[{"to":"deriv-sub","from":"deriv","color":"amber","label":"rearrange","style":"solid"}]},"order":1,"title":"Define the substitution","description":"We define a new variable $u$ to simplify the exponent of the exponential function."},{"type":"text","order":2,"title":"Substitute into the integral","content":"Now we substitute both $u = 2x + 3$ and $dx = \\frac{1}{2} \\, du$ back into the original integral:\n\n$$\\int e^{2x+3} \\, dx = \\int e^{u} \\left( \\frac{1}{2} \\, du \\right)$$\n\nBy pulling the constant factor $\\frac{1}{2}$ out of the integration sign, we get:\n\n$$\\frac{1}{2} \\int e^{u} \\, du$$"},{"type":"text","order":3,"title":"Integrate with respect to u","content":"The integral of $e^u$ is a standard integral from our formula booklet:\n\n$$\\int e^u \\, du = e^u + C$$\n\nApplying this standard form, we get:\n\n$$\\frac{1}{2} e^u + C$$"},{"type":"text","order":4,"title":"Substitute back to x","content":"Finally, we substitute $u = 2x+3$ back into our result to express our final answer in terms of $x$:\n\n$$\\frac{1}{2} e^{2x+3} + C$$\n\n### Markscheme Breakdown\n- **M1**: Integrating the exponential function to get the correct general form.\n- **A1**: Finding the correct coefficient $\\frac{1}{2}$.\n- **B1**: Including the constant of integration $+ C$."}]}],"generatedAt":"2026-06-09T03:33:53.819Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"0f13d61c-9b51-4ddd-8a76-5b68c239d9fd","revisionId":"af868bc5-39dc-467c-a11a-f95f6eae4334","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"64366fa3-eba8-4c8b-a669-6b9bea4eeeb3","label":"a","content":"Use the substitution $u = x^3 + 1$ to find $\\int 3x^2(x^3 + 1)^3 dx$","markscheme":"$$u = x^3 + 1 \\implies \\frac{du}{dx} = 3x^2$ M1\n$\\int 3x^2(u)^3 \\frac{1}{3x^2} du = \\int u^3 du$ M1\n$= \\frac{1}{4}u^4 + c$ M1\n$= \\frac{1}{4}(x^3 + 1)^4 + c$ A2","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Differentiates the substitution $u = x^3 + 1$ to find $\\frac{du}{dx}$","code":"M"},{"id":"p2","marks":1,"description":"Correctly substitutes $u$ and $dx$ into the integral to eliminate $x$","code":"M"},{"id":"p3","marks":1,"description":"Integrates the resulting power of $u$","code":"M"},{"id":"p4","marks":2,"description":"Replaces $u$ with the original expression in $x$ and includes the constant of integration","code":"A"}],"finalAnswers":["\\frac{1}{4}(x^3 + 1)^4 + c"],"generatedAt":"2026-05-06T14:04:28.838Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"dx_substitution","label":"Substitution by Isolating dx","steps":[{"type":"text","order":0,"title":"Differentiate the substitution","content":"First, we start with the given substitution:\n\n$$u = x^3 + 1$$\n\nDifferentiate $u$ with respect to $x$ to find $\\frac{du}{dx}$:\n\n$$\\frac{du}{dx} = 3x^2$$\n\nNow, rearrange this equation to isolate $dx$:\n\n$$dx = \\frac{du}{3x^2}$$\n\n*Awarded 1 mark (M1) for finding $\\frac{du}{dx} = 3x^2$.*","highlights":[{"text":"substitution u = x^3 + 1","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int 3x^2 \\left(\\htmlData{anchor=u-sub}{\\color{@sky}{x^3 + 1}}\\right)^3 \\htmlData{anchor=dx-sub}{\\color{@amber}{dx}}$"},{"latex":"$$\\int \\htmlData{anchor=x-term}{\\color{@pink}{3x^2}} \\left(\\color{@sky}{u}\\right)^3 \\left(\\frac{du}{\\htmlData{anchor=dx-expr}{\\color{@pink}{3x^2}}}\\right)$"},{"latex":"$$\\int u^3 \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-sub"},{"color":"amber","style":"text-color","anchor":"dx-sub"}],"connections":[{"to":"dx-sub","from":"u-sub","color":"sky","label":"substitute u and dx","style":"solid"},{"to":"dx-expr","from":"x-term","color":"pink","label":"cancel","style":"dashed"}]},"order":1,"title":"Substitute and simplify","description":"Substitute $u$ and $dx = \\frac{du}{3x^2}$ into our original integral. This lets us cancel the $3x^2$ term outside. This earns the second mark (M1)."},{"type":"text","order":2,"title":"Integrate with respect to u","content":"Now we integrate $u^3$ with respect to $u$ using the power rule:\n\n$$\\int u^3 \\, du = \\frac{u^4}{4} + c$$\n\n*Awarded 1 mark (M1) for correctly integrating to get $\\frac{1}{4}u^4 + c$.*"},{"type":"text","order":3,"title":"Substitute back to x","content":"Finally, replace $u$ with the original expression $x^3 + 1$ to get the final answer:\n\n$$\\frac{1}{4}(x^3 + 1)^4 + c$$\n\n*Awarded 2 marks (A2) for the correct final expression including the constant of integration $c$.*"}]},{"id":"differential_substitution","label":"Direct Differential Substitution","steps":[{"type":"text","order":0,"title":"Identify the substitution","content":"We are given the substitution:\n\n$$u = x^3 + 1$$\n\nLet's find the derivative of $u$ with respect to $x$:\n\n$$\\frac{du}{dx} = 3x^2$$","highlights":[{"text":"substitution u = x^3 + 1","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Since } \\frac{du}{dx} = 3x^2, \\text{ we can write the differential relationship directly as:}$"},{"latex":"$$\\htmlData{anchor=du}{\\color{@sky}{du}} = \\htmlData{anchor=diff}{\\color{@sky}{3x^2 dx}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"du"},{"color":"sky","style":"text-color","anchor":"diff"}],"connections":[]},"order":1,"title":"Find the differential","description":"Writing the differential relation explicitly allows us to identify and substitute components of our integrand directly."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Let's rearrange the original integral to group the terms:}$"},{"latex":"$$\\int (\\htmlData{anchor=u-part}{\\color{@amber}{x^3 + 1}})^3 \\cdot (\\htmlData{anchor=du-part}{\\color{@sky}{3x^2 dx}})$"},{"latex":"$$\\text{Now, substitute } \\htmlData{anchor=u-sub}{\\color{@amber}{u = x^3 + 1}} \\text{ and } \\htmlData{anchor=du-sub}{\\color{@sky}{du = 3x^2 dx}} \\text{:}$"},{"latex":"$$\\int \\htmlData{anchor=u-expr}{\\color{@amber}{u}}^3 \\htmlData{anchor=du-expr}{\\color{@sky}{du}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"u-part"},{"color":"sky","style":"text-color","anchor":"du-part"},{"color":"amber","style":"text-color","anchor":"u-expr"},{"color":"sky","style":"text-color","anchor":"du-expr"}],"connections":[{"to":"u-sub","from":"u-part","color":"amber","label":"Substitute u","style":"solid"},{"to":"du-sub","from":"du-part","color":"sky","label":"Substitute du","style":"solid"}]},"order":2,"title":"Perform direct substitution","description":"By grouping $3x^2$ with $dx$, we can directly substitute $du$ and simplify the integral."},{"type":"text","order":3,"title":"Integrate with respect to u","content":"Now we integrate $u^3$ with respect to $u$ using the power rule:\n\n$$\\int u^3 du = \\frac{1}{4}u^4 + c$$\n\nwhere $c$ is the constant of integration."},{"type":"text","order":4,"title":"Substitute back for x","content":"Finally, substitute the original expression for $u$ back into the result to write the answer in terms of $x$:\n\nSince $u = x^3 + 1$, we have:\n\n$$\\frac{1}{4}(x^3 + 1)^4 + c$$"}]}],"generatedAt":"2026-06-09T03:18:33.546Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"64366fa3-eba8-4c8b-a669-6b9bea4eeeb3:dx_substitution","label":"Substitution by Isolating dx","steps":[{"type":"text","order":0,"title":"Differentiate the substitution","content":"First, we start with the given substitution:\n\n$$u = x^3 + 1$$\n\nDifferentiate $u$ with respect to $x$ to find $\\frac{du}{dx}$:\n\n$$\\frac{du}{dx} = 3x^2$$\n\nNow, rearrange this equation to isolate $dx$:\n\n$$dx = \\frac{du}{3x^2}$$\n\n*Awarded 1 mark (M1) for finding $\\frac{du}{dx} = 3x^2$.*","highlights":[{"text":"substitution u = x^3 + 1","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int 3x^2 \\left(\\htmlData{anchor=u-sub}{\\color{@sky}{x^3 + 1}}\\right)^3 \\htmlData{anchor=dx-sub}{\\color{@amber}{dx}}$"},{"latex":"$$\\int \\htmlData{anchor=x-term}{\\color{@pink}{3x^2}} \\left(\\color{@sky}{u}\\right)^3 \\left(\\frac{du}{\\htmlData{anchor=dx-expr}{\\color{@pink}{3x^2}}}\\right)$"},{"latex":"$$\\int u^3 \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-sub"},{"color":"amber","style":"text-color","anchor":"dx-sub"}],"connections":[{"to":"dx-sub","from":"u-sub","color":"sky","label":"substitute u and dx","style":"solid"},{"to":"dx-expr","from":"x-term","color":"pink","label":"cancel","style":"dashed"}]},"order":1,"title":"Substitute and simplify","description":"Substitute $u$ and $dx = \\frac{du}{3x^2}$ into our original integral. This lets us cancel the $3x^2$ term outside. This earns the second mark (M1)."},{"type":"text","order":2,"title":"Integrate with respect to u","content":"Now we integrate $u^3$ with respect to $u$ using the power rule:\n\n$$\\int u^3 \\, du = \\frac{u^4}{4} + c$$\n\n*Awarded 1 mark (M1) for correctly integrating to get $\\frac{1}{4}u^4 + c$.*"},{"type":"text","order":3,"title":"Substitute back to x","content":"Finally, replace $u$ with the original expression $x^3 + 1$ to get the final answer:\n\n$$\\frac{1}{4}(x^3 + 1)^4 + c$$\n\n*Awarded 2 marks (A2) for the correct final expression including the constant of integration $c$.*"}]},{"id":"64366fa3-eba8-4c8b-a669-6b9bea4eeeb3:differential_substitution","label":"Direct Differential Substitution","steps":[{"type":"text","order":0,"title":"Identify the substitution","content":"We are given the substitution:\n\n$$u = x^3 + 1$$\n\nLet's find the derivative of $u$ with respect to $x$:\n\n$$\\frac{du}{dx} = 3x^2$$","highlights":[{"text":"substitution u = x^3 + 1","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Since } \\frac{du}{dx} = 3x^2, \\text{ we can write the differential relationship directly as:}$"},{"latex":"$$\\htmlData{anchor=du}{\\color{@sky}{du}} = \\htmlData{anchor=diff}{\\color{@sky}{3x^2 dx}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"du"},{"color":"sky","style":"text-color","anchor":"diff"}],"connections":[]},"order":1,"title":"Find the differential","description":"Writing the differential relation explicitly allows us to identify and substitute components of our integrand directly."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Let's rearrange the original integral to group the terms:}$"},{"latex":"$$\\int (\\htmlData{anchor=u-part}{\\color{@amber}{x^3 + 1}})^3 \\cdot (\\htmlData{anchor=du-part}{\\color{@sky}{3x^2 dx}})$"},{"latex":"$$\\text{Now, substitute } \\htmlData{anchor=u-sub}{\\color{@amber}{u = x^3 + 1}} \\text{ and } \\htmlData{anchor=du-sub}{\\color{@sky}{du = 3x^2 dx}} \\text{:}$"},{"latex":"$$\\int \\htmlData{anchor=u-expr}{\\color{@amber}{u}}^3 \\htmlData{anchor=du-expr}{\\color{@sky}{du}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"u-part"},{"color":"sky","style":"text-color","anchor":"du-part"},{"color":"amber","style":"text-color","anchor":"u-expr"},{"color":"sky","style":"text-color","anchor":"du-expr"}],"connections":[{"to":"u-sub","from":"u-part","color":"amber","label":"Substitute u","style":"solid"},{"to":"du-sub","from":"du-part","color":"sky","label":"Substitute du","style":"solid"}]},"order":2,"title":"Perform direct substitution","description":"By grouping $3x^2$ with $dx$, we can directly substitute $du$ and simplify the integral."},{"type":"text","order":3,"title":"Integrate with respect to u","content":"Now we integrate $u^3$ with respect to $u$ using the power rule:\n\n$$\\int u^3 du = \\frac{1}{4}u^4 + c$$\n\nwhere $c$ is the constant of integration."},{"type":"text","order":4,"title":"Substitute back for x","content":"Finally, substitute the original expression for $u$ back into the result to write the answer in terms of $x$:\n\nSince $u = x^3 + 1$, we have:\n\n$$\\frac{1}{4}(x^3 + 1)^4 + c$$"}]}],"generatedAt":"2026-06-09T03:18:56.469Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"3eacbbf9-f07a-4d1f-8856-766abf8a4f98","revisionId":"d5903e80-90c9-42fc-a7f6-71ec0f546cab","questionNumber":null,"questionType":"LA","specification":"Integrate with respect to $x$:","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"e442c368-4e39-4f79-bfa4-4eee41ab1a46","label":"a","content":"$$\\sqrt{2x - 7}$","markscheme":"- Rewrites as $(2x-7)^{\\frac{1}{2}}$ and integrates to the form $k(2x-7)^{\\frac{3}{2}}$ M1\n- Simplifies to $\\frac{1}{3}(2x-7)^{\\frac{3}{2}}$ A1\n- Includes constant of integration $+ C$ B1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Attempt to integrate power to obtain $(2x-7)^{\\frac{3}{2}}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct coefficient $\\frac{1}{3}$ or equivalent (from $\\frac{2}{3} \\times \\frac{1}{2}$).","code":"A"},{"id":"p3","marks":1,"description":"Include $+ C$ in the final answer.","code":"B"}],"finalAnswers":["\\frac{1}{3}(2x-7)^{\\frac{3}{2}} + C"],"generatedAt":"2026-05-06T14:04:09.699Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"reverse_chain_rule","label":"Reverse Chain Rule","steps":[{"type":"text","order":0,"title":"Rewrite using index form","content":"To integrate $\\sqrt{2x - 7}$ with respect to $x$, we first rewrite the square root in index (power) form:\n\n$$\\int \\sqrt{2x - 7} \\, dx = \\int (2x - 7)^{\\frac{1}{2}} \\, dx$$","highlights":[{"text":"$$\\sqrt{2x - 7}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$a = \\htmlData{anchor=val-a}{\\color{@sky}{2}}, \\quad n = \\htmlData{anchor=val-n}{\\color{@purple}{\\frac{1}{2}}}$$"},{"latex":"$$$\\int (\\htmlData{anchor=base-form}{ax+b})^n \\, dx = \\frac{(\\htmlData{anchor=base-res}{ax+b})^{n+1}}{\\htmlData{anchor=rule-a}{a}(\\htmlData{anchor=rule-n}{n+1})} + C$$"},{"latex":"$$$\\int (2x-7)^{\\frac{1}{2}} \\, dx = \\frac{(2x-7)^{\\frac{1}{2}+1}}{\\htmlData{anchor=sub-a}{\\color{@sky}{2}}\\left(\\htmlData{anchor=sub-n}{\\color{@purple}{\\frac{1}{2}}}+1\\right)} + C$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"val-a"},{"color":"purple","style":"text-color","anchor":"val-n"},{"color":"sky","style":"text-color","anchor":"sub-a"},{"color":"purple","style":"text-color","anchor":"sub-n"}],"connections":[{"to":"sub-a","from":"val-a","color":"sky","label":null,"style":"solid"},{"to":"sub-n","from":"val-n","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Apply Reverse Chain Rule","description":"We apply the general integration rule for linear composite functions: \n\n$$\\int (ax+b)^n \\, dx = \\frac{(ax+b)^{n+1}}{a(n+1)} + C$$\n\nHere, the inner function is $2x - 7$ (which has $a = 2$) and the exponent is $n = \\frac{1}{2}$."},{"type":"text","order":2,"title":"Simplify the expression","content":"Now we simplify the power and the denominator:\n\n* **Power:** $\\frac{1}{2} + 1 = \\frac{3}{2}$\n* **Denominator:** $2 \\times \\left(\\frac{1}{2} + 1\\right) = 2 \\times \\frac{3}{2} = 3$\n\nSubstituting these values back into our integration step yields:\n\n$$\\int (2x-7)^{\\frac{1}{2}} \\, dx = \\frac{(2x-7)^{\\frac{3}{2}}}{3} + C$$\n\nWhich can be written as:\n\n$$\\frac{1}{3}(2x-7)^{\\frac{3}{2}} + C$$"},{"type":"text","order":3,"title":"State final mark breakdown","content":"Our final answer is:\n\n$$\\frac{1}{3}(2x-7)^{\\frac{3}{2}} + C$$\n\n**Marks awarded:**\n* **M1**: Rewriting the root as $(2x-7)^{\\frac{1}{2}}$ and integrating to the form $k(2x-7)^{\\frac{3}{2}}$\n* **A1**: Correctly simplifying the coefficient to $\\frac{1}{3}(2x-7)^{\\frac{3}{2}}$\n* **B1**: Including the constant of integration $+ C$"}]},{"id":"substitution","label":"Integration by Substitution","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We want to find the indefinite integral of the given function with respect to $x$:\n\n$$\\int \\sqrt{2x - 7} \\, dx$$\n\nTo make this easier to integrate, we can rewrite the radical using a fractional exponent:\n\n$$\\int (2x - 7)^{\\frac{1}{2}} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"u = \\htmlData{anchor=sub-u}{2x - 7}"},{"text":"Differentiating both sides with respect to $x$ gives:"},{"latex":"\\frac{du}{dx} = 2"},{"text":"Which we can rewrite to find an expression for $dx$:"},{"latex":"dx = \\htmlData{anchor=sub-dx}{\\frac{1}{2} \\, du}"}],"highlights":[{"color":"sky","style":"box","anchor":"sub-u"},{"color":"amber","style":"box","anchor":"sub-dx"}],"connections":[]},"order":1,"title":"Define the substitution","description":"We choose $u = 2x - 7$ to simplify the inner function. Then we find $dx$ in terms of $du$ so we can substitute it into our integral."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\int (\\htmlData{anchor=int-u}{\\color{@sky}{2x - 7}})^{\\frac{1}{2}} \\, \\htmlData{anchor=int-dx}{\\color{@amber}{dx}}"},{"latex":"= \\int (\\htmlData{anchor=res-u}{\\color{@sky}{u}})^{\\frac{1}{2}} \\left( \\htmlData{anchor=res-dx}{\\color{@amber}{\\frac{1}{2} \\, du}} \\right)"},{"latex":"= \\frac{1}{2} \\int u^{\\frac{1}{2}} \\, du"}],"highlights":[{"color":"sky","style":"text-color","anchor":"int-u"},{"color":"sky","style":"text-color","anchor":"res-u"},{"color":"amber","style":"text-color","anchor":"int-dx"},{"color":"amber","style":"text-color","anchor":"res-dx"}],"connections":[{"to":"res-u","from":"int-u","color":"sky","label":"substitute u","style":"solid"},{"to":"res-dx","from":"int-dx","color":"amber","label":"substitute dx","style":"solid"}]},"order":2,"title":"Substitute into the integral","description":"Now we replace $2x - 7$ with $u$, and $dx$ with $\\frac{1}{2} \\, du$ in our integral."},{"type":"text","order":3,"title":"Integrate using power rule","content":"Now we integrate $u^{\\frac{1}{2}}$ with respect to $u$ using the power rule, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1}$ (for $n \\neq -1$):\n\n$$\\begin{aligned} \\frac{1}{2} \\int u^{\\frac{1}{2}} \\, du &= \\frac{1}{2} \\left( \\frac{u^{\\frac{3}{2}}}{\\frac{3}{2}} \\right) + C \\\\ &= \\frac{1}{2} \\cdot \\frac{2}{3} u^{\\frac{3}{2}} + C \\\\ &= \\frac{1}{3} u^{\\frac{3}{2}} + C \\end{aligned}$$\n\n*Note: This earns us the **M1** mark for obtaining the form $k(2x-7)^{\\frac{3}{2}}$ and the **B1** mark for including the constant of integration $+ C$.*"},{"type":"text","order":4,"title":"Substitute back to x","content":"Finally, we substitute our original definition $u = 2x - 7$ back into the expression:\n\n$$\\frac{1}{3}(2x - 7)^{\\frac{3}{2}} + C$$\n\nThis gives us our final answer, earning the final **A1** mark."}]}],"generatedAt":"2026-06-09T04:45:44.374Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"192e5e23-2d30-4e54-b428-4cb49fa5b683","label":"b","content":"$$e^{4x + 2}$","markscheme":"- Integrates exponential to the form $k e^{4x+2}$ M1\n- Correct coefficient $\\frac{1}{4}$ A1\n- Includes constant of integration $+ C$ B1","marks":3,"order":1,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Integrate to obtain $k e^{4x+2}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct coefficient $\\frac{1}{4}$.","code":"A"},{"id":"p3","marks":1,"description":"Include $+ C$ in the final answer.","code":"B"}],"finalAnswers":["\\frac{1}{4}e^{4x+2} + C"],"generatedAt":"2026-05-06T14:04:09.699Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"substitution","label":"Integration by Substitution","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We want to integrate the given function $e^{4x + 2}$ with respect to $x$:\n\n$$\\int e^{4x+2} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int e^{\\htmlData{anchor=orig-exp}{\\color{@sky}{4x+2}}} \\, \\htmlData{anchor=orig-dx}{\\color{@amber}{dx}}$"},{"latex":"$$\\text{Let } \\htmlData{anchor=sub-u}{\\color{@sky}{u}} = 4x + 2 \\implies \\frac{du}{dx} = 4 \\implies \\htmlData{anchor=sub-dx}{\\color{@amber}{dx}} = \\htmlData{anchor=sub-du}{\\color{@purple}{\\frac{1}{4} du}}$"},{"latex":"$$=\\int e^{\\htmlData{anchor=new-u}{\\color{@sky}{u}}} \\, \\htmlData{anchor=new-dx}{\\color{@purple}{\\frac{1}{4} du}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"orig-exp"},{"color":"sky","style":"text-color","anchor":"sub-u"},{"color":"sky","style":"text-color","anchor":"new-u"},{"color":"purple","style":"text-color","anchor":"sub-du"},{"color":"purple","style":"text-color","anchor":"new-dx"}],"connections":[{"to":"sub-u","from":"orig-exp","color":"sky","label":"Substitute exponent","style":"solid"},{"to":"new-u","from":"sub-u","color":"sky","label":"New variable","style":"solid"},{"to":"sub-dx","from":"orig-dx","color":"amber","label":"Differential term","style":"solid"},{"to":"new-dx","from":"sub-du","color":"purple","label":"Substitute dx","style":"solid"}]},"order":1,"title":"Apply substitution","description":"We choose $u = 4x+2$ to simplify the exponent, then find the differential $dx$ in terms of $du$ to perform the complete change of variables."},{"type":"text","order":2,"title":"Integrate with respect to u","content":"Factor out the constant $\\frac{1}{4}$ and integrate the exponential function:\n\n$$\\int e^u \\left(\\frac{1}{4} du\\right) = \\frac{1}{4} \\int e^u \\, du$$\n\nSince the integral of $e^u$ is simply $e^u$:\n\n$$= \\frac{1}{4} e^u + C$$"},{"type":"text","order":3,"title":"Substitute back to x","content":"Finally, replace $u$ with the original expression $4x + 2$ to get our final answer:\n\n$$\\frac{1}{4} e^{4x+2} + C$$"}]},{"id":"inspection","label":"Direct Integration","steps":[{"type":"text","order":0,"title":"Identify the standard rule","content":"To integrate an exponential function where the exponent is a linear function of the form $ax + b$, we can use the direct integration rule:\n\n$$\\int e^{ax+b} \\, dx = \\frac{1}{a} e^{ax+b} + C$$\n\nFor this problem, we want to find:\n\n$$\\int e^{4x + 2} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int e^{\\htmlData{anchor=rule-a}{\\color{@sky}{a}}x+b} \\, dx = \\frac{1}{\\htmlData{anchor=rule-recip}{\\color{@sky}{a}}}e^{ax+b} + C$"},{"latex":"$$\\int e^{\\htmlData{anchor=val-a}{\\color{@sky}{4}}x+2} \\, dx = \\frac{1}{\\htmlData{anchor=val-recip}{\\color{@sky}{4}}}e^{4x+2} + C$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"rule-a"},{"color":"sky","style":"text-color","anchor":"val-a"},{"color":"sky","style":"text-color","anchor":"rule-recip"},{"color":"sky","style":"text-color","anchor":"val-recip"}],"connections":[{"to":"val-a","from":"rule-a","color":"sky","label":"a = 4","style":"solid"},{"to":"val-recip","from":"rule-recip","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Apply standard formula","description":"We identify that the coefficient of $x$ is $a = 4$ and apply the integration rule directly."},{"type":"text","order":2,"title":"Write final answer","content":"Thus, the result of the integration is:\n\n$$\\frac{1}{4}e^{4x + 2} + C$$\n\n### Markscheme Breakdown\n- **M1**: Integrating to the form $k e^{4x+2}$ where $k$ is a constant.\n- **A1**: Finding the correct coefficient $k = \\frac{1}{4}$.\n- **B1**: Including the constant of integration $+ C$."}]}],"generatedAt":"2026-06-09T04:47:36.154Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"e442c368-4e39-4f79-bfa4-4eee41ab1a46:reverse_chain_rule","label":"(a) Reverse Chain Rule","steps":[{"type":"text","order":0,"title":"Rewrite using index form","content":"To integrate $\\sqrt{2x - 7}$ with respect to $x$, we first rewrite the square root in index (power) form:\n\n$$\\int \\sqrt{2x - 7} \\, dx = \\int (2x - 7)^{\\frac{1}{2}} \\, dx$$","highlights":[{"text":"$$\\sqrt{2x - 7}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$a = \\htmlData{anchor=val-a}{\\color{@sky}{2}}, \\quad n = \\htmlData{anchor=val-n}{\\color{@purple}{\\frac{1}{2}}}$$"},{"latex":"$$$\\int (\\htmlData{anchor=base-form}{ax+b})^n \\, dx = \\frac{(\\htmlData{anchor=base-res}{ax+b})^{n+1}}{\\htmlData{anchor=rule-a}{a}(\\htmlData{anchor=rule-n}{n+1})} + C$$"},{"latex":"$$$\\int (2x-7)^{\\frac{1}{2}} \\, dx = \\frac{(2x-7)^{\\frac{1}{2}+1}}{\\htmlData{anchor=sub-a}{\\color{@sky}{2}}\\left(\\htmlData{anchor=sub-n}{\\color{@purple}{\\frac{1}{2}}}+1\\right)} + C$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"val-a"},{"color":"purple","style":"text-color","anchor":"val-n"},{"color":"sky","style":"text-color","anchor":"sub-a"},{"color":"purple","style":"text-color","anchor":"sub-n"}],"connections":[{"to":"sub-a","from":"val-a","color":"sky","label":null,"style":"solid"},{"to":"sub-n","from":"val-n","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Apply Reverse Chain Rule","description":"We apply the general integration rule for linear composite functions: \n\n$$\\int (ax+b)^n \\, dx = \\frac{(ax+b)^{n+1}}{a(n+1)} + C$$\n\nHere, the inner function is $2x - 7$ (which has $a = 2$) and the exponent is $n = \\frac{1}{2}$."},{"type":"text","order":2,"title":"Simplify the expression","content":"Now we simplify the power and the denominator:\n\n* **Power:** $\\frac{1}{2} + 1 = \\frac{3}{2}$\n* **Denominator:** $2 \\times \\left(\\frac{1}{2} + 1\\right) = 2 \\times \\frac{3}{2} = 3$\n\nSubstituting these values back into our integration step yields:\n\n$$\\int (2x-7)^{\\frac{1}{2}} \\, dx = \\frac{(2x-7)^{\\frac{3}{2}}}{3} + C$$\n\nWhich can be written as:\n\n$$\\frac{1}{3}(2x-7)^{\\frac{3}{2}} + C$$"},{"type":"text","order":3,"title":"State final mark breakdown","content":"Our final answer is:\n\n$$\\frac{1}{3}(2x-7)^{\\frac{3}{2}} + C$$\n\n**Marks awarded:**\n* **M1**: Rewriting the root as $(2x-7)^{\\frac{1}{2}}$ and integrating to the form $k(2x-7)^{\\frac{3}{2}}$\n* **A1**: Correctly simplifying the coefficient to $\\frac{1}{3}(2x-7)^{\\frac{3}{2}}$\n* **B1**: Including the constant of integration $+ C$"}]},{"id":"e442c368-4e39-4f79-bfa4-4eee41ab1a46:substitution","label":"(a) Integration by Substitution","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We want to find the indefinite integral of the given function with respect to $x$:\n\n$$\\int \\sqrt{2x - 7} \\, dx$$\n\nTo make this easier to integrate, we can rewrite the radical using a fractional exponent:\n\n$$\\int (2x - 7)^{\\frac{1}{2}} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"u = \\htmlData{anchor=sub-u}{2x - 7}"},{"text":"Differentiating both sides with respect to $x$ gives:"},{"latex":"\\frac{du}{dx} = 2"},{"text":"Which we can rewrite to find an expression for $dx$:"},{"latex":"dx = \\htmlData{anchor=sub-dx}{\\frac{1}{2} \\, du}"}],"highlights":[{"color":"sky","style":"box","anchor":"sub-u"},{"color":"amber","style":"box","anchor":"sub-dx"}],"connections":[]},"order":1,"title":"Define the substitution","description":"We choose $u = 2x - 7$ to simplify the inner function. Then we find $dx$ in terms of $du$ so we can substitute it into our integral."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\int (\\htmlData{anchor=int-u}{\\color{@sky}{2x - 7}})^{\\frac{1}{2}} \\, \\htmlData{anchor=int-dx}{\\color{@amber}{dx}}"},{"latex":"= \\int (\\htmlData{anchor=res-u}{\\color{@sky}{u}})^{\\frac{1}{2}} \\left( \\htmlData{anchor=res-dx}{\\color{@amber}{\\frac{1}{2} \\, du}} \\right)"},{"latex":"= \\frac{1}{2} \\int u^{\\frac{1}{2}} \\, du"}],"highlights":[{"color":"sky","style":"text-color","anchor":"int-u"},{"color":"sky","style":"text-color","anchor":"res-u"},{"color":"amber","style":"text-color","anchor":"int-dx"},{"color":"amber","style":"text-color","anchor":"res-dx"}],"connections":[{"to":"res-u","from":"int-u","color":"sky","label":"substitute u","style":"solid"},{"to":"res-dx","from":"int-dx","color":"amber","label":"substitute dx","style":"solid"}]},"order":2,"title":"Substitute into the integral","description":"Now we replace $2x - 7$ with $u$, and $dx$ with $\\frac{1}{2} \\, du$ in our integral."},{"type":"text","order":3,"title":"Integrate using power rule","content":"Now we integrate $u^{\\frac{1}{2}}$ with respect to $u$ using the power rule, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1}$ (for $n \\neq -1$):\n\n$$\\begin{aligned} \\frac{1}{2} \\int u^{\\frac{1}{2}} \\, du &= \\frac{1}{2} \\left( \\frac{u^{\\frac{3}{2}}}{\\frac{3}{2}} \\right) + C \\\\ &= \\frac{1}{2} \\cdot \\frac{2}{3} u^{\\frac{3}{2}} + C \\\\ &= \\frac{1}{3} u^{\\frac{3}{2}} + C \\end{aligned}$$\n\n*Note: This earns us the **M1** mark for obtaining the form $k(2x-7)^{\\frac{3}{2}}$ and the **B1** mark for including the constant of integration $+ C$.*"},{"type":"text","order":4,"title":"Substitute back to x","content":"Finally, we substitute our original definition $u = 2x - 7$ back into the expression:\n\n$$\\frac{1}{3}(2x - 7)^{\\frac{3}{2}} + C$$\n\nThis gives us our final answer, earning the final **A1** mark."}]},{"id":"192e5e23-2d30-4e54-b428-4cb49fa5b683:substitution","label":"(b) Integration by Substitution","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We want to integrate the given function $e^{4x + 2}$ with respect to $x$:\n\n$$\\int e^{4x+2} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int e^{\\htmlData{anchor=orig-exp}{\\color{@sky}{4x+2}}} \\, \\htmlData{anchor=orig-dx}{\\color{@amber}{dx}}$"},{"latex":"$$\\text{Let } \\htmlData{anchor=sub-u}{\\color{@sky}{u}} = 4x + 2 \\implies \\frac{du}{dx} = 4 \\implies \\htmlData{anchor=sub-dx}{\\color{@amber}{dx}} = \\htmlData{anchor=sub-du}{\\color{@purple}{\\frac{1}{4} du}}$"},{"latex":"$$=\\int e^{\\htmlData{anchor=new-u}{\\color{@sky}{u}}} \\, \\htmlData{anchor=new-dx}{\\color{@purple}{\\frac{1}{4} du}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"orig-exp"},{"color":"sky","style":"text-color","anchor":"sub-u"},{"color":"sky","style":"text-color","anchor":"new-u"},{"color":"purple","style":"text-color","anchor":"sub-du"},{"color":"purple","style":"text-color","anchor":"new-dx"}],"connections":[{"to":"sub-u","from":"orig-exp","color":"sky","label":"Substitute exponent","style":"solid"},{"to":"new-u","from":"sub-u","color":"sky","label":"New variable","style":"solid"},{"to":"sub-dx","from":"orig-dx","color":"amber","label":"Differential term","style":"solid"},{"to":"new-dx","from":"sub-du","color":"purple","label":"Substitute dx","style":"solid"}]},"order":1,"title":"Apply substitution","description":"We choose $u = 4x+2$ to simplify the exponent, then find the differential $dx$ in terms of $du$ to perform the complete change of variables."},{"type":"text","order":2,"title":"Integrate with respect to u","content":"Factor out the constant $\\frac{1}{4}$ and integrate the exponential function:\n\n$$\\int e^u \\left(\\frac{1}{4} du\\right) = \\frac{1}{4} \\int e^u \\, du$$\n\nSince the integral of $e^u$ is simply $e^u$:\n\n$$= \\frac{1}{4} e^u + C$$"},{"type":"text","order":3,"title":"Substitute back to x","content":"Finally, replace $u$ with the original expression $4x + 2$ to get our final answer:\n\n$$\\frac{1}{4} e^{4x+2} + C$$"}]},{"id":"192e5e23-2d30-4e54-b428-4cb49fa5b683:inspection","label":"(b) Direct Integration","steps":[{"type":"text","order":0,"title":"Identify the standard rule","content":"To integrate an exponential function where the exponent is a linear function of the form $ax + b$, we can use the direct integration rule:\n\n$$\\int e^{ax+b} \\, dx = \\frac{1}{a} e^{ax+b} + C$$\n\nFor this problem, we want to find:\n\n$$\\int e^{4x + 2} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int e^{\\htmlData{anchor=rule-a}{\\color{@sky}{a}}x+b} \\, dx = \\frac{1}{\\htmlData{anchor=rule-recip}{\\color{@sky}{a}}}e^{ax+b} + C$"},{"latex":"$$\\int e^{\\htmlData{anchor=val-a}{\\color{@sky}{4}}x+2} \\, dx = \\frac{1}{\\htmlData{anchor=val-recip}{\\color{@sky}{4}}}e^{4x+2} + C$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"rule-a"},{"color":"sky","style":"text-color","anchor":"val-a"},{"color":"sky","style":"text-color","anchor":"rule-recip"},{"color":"sky","style":"text-color","anchor":"val-recip"}],"connections":[{"to":"val-a","from":"rule-a","color":"sky","label":"a = 4","style":"solid"},{"to":"val-recip","from":"rule-recip","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Apply standard formula","description":"We identify that the coefficient of $x$ is $a = 4$ and apply the integration rule directly."},{"type":"text","order":2,"title":"Write final answer","content":"Thus, the result of the integration is:\n\n$$\\frac{1}{4}e^{4x + 2} + C$$\n\n### Markscheme Breakdown\n- **M1**: Integrating to the form $k e^{4x+2}$ where $k$ is a constant.\n- **A1**: Finding the correct coefficient $k = \\frac{1}{4}$.\n- **B1**: Including the constant of integration $+ 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1.8.8 Integration by substitution (A-level only)

Question 5

1.8.8 Integration by substitution (A-level only) Questions

1.8.8 Integration by substitution (A-level only)

Question 5

1.8.8 Integration by substitution (A-level only) Questions