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B1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Evidence of integrating to the form $ke^{2x+3}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct coefficient of $\\frac{1}{2}$.","code":"A"},{"id":"p3","marks":1,"description":"Includes constant of integration $+ C$.","code":"B"}],"finalAnswers":["\\frac{1}{2}e^{2x+3} + C"],"generatedAt":"2026-05-06T14:04:13.116Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"reverse-chain-rule","label":"Reverse Chain Rule","steps":[{"type":"text","order":0,"title":"Identify the standard rule","content":"To integrate an exponential function of the form $e^{ax+b}$, we can use the reverse chain rule.\n\nRecall that differentiating $e^{ax+b}$ with respect to $x$ yields $a e^{ax+b}$ by the chain rule. Therefore, when integrating, we do the reverse by dividing by the coefficient of $x$:\n\n$$\\int e^{ax+b} \\, dx = \\frac{1}{a} e^{ax+b} + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$e^{\\htmlData{anchor=a-val}{\\color{@sky}{2}}x + \\htmlData{anchor=b-val}{\\color{@amber}{3}}}$"},{"latex":"$$a = \\htmlData{anchor=a-sub}{\\color{@sky}{2}}, \\quad b = \\htmlData{anchor=b-sub}{\\color{@amber}{3}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-val"},{"color":"amber","style":"text-color","anchor":"b-val"}],"connections":[{"to":"a-sub","from":"a-val","color":"sky","label":null,"style":"solid"},{"to":"b-sub","from":"b-val","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Identify coefficients","description":"In our expression $e^{2x+3}$, we identify the coefficient of $x$ as $a = 2$ and the constant term as $b = 3$."},{"type":"text","order":2,"title":"Apply reverse chain rule","content":"Using our formula with $a = 2$:\n\n$$\\int e^{2x+3} \\, dx = \\frac{1}{2}e^{2x+3} + C$$\n\nThis step earns you the method mark (**M1**) for integrating the exponential and the accuracy mark (**A1**) for the correct coefficient of $\\frac{1}{2}$."},{"type":"text","order":3,"title":"Include integration constant","content":"Don't forget to include the constant of integration, $+ C$!\n\nThis final detail completes the answer and earns the remaining mark (**B1**):\n\n$$\\frac{1}{2}e^{2x+3} + C$$"}]},{"id":"substitution","label":"Integration by Substitution","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We want to integrate the given expression with respect to $x$:\n\n$$\\int e^{2x+3} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = \\htmlData{anchor=sub-u}{\\color{@sky}{2x + 3}}$"},{"text":"Now, we differentiate $u$ with respect to $x$ to find the relationship between $du$ and $dx$:"},{"latex":"$$\\frac{du}{dx} = \\htmlData{anchor=diff-u}{\\color{@amber}{2}} \\implies dx = \\htmlData{anchor=sub-dx}{\\color{@purple}{\\frac{1}{2}}} \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-u"},{"color":"purple","style":"text-color","anchor":"sub-dx"}],"connections":[{"to":"diff-u","from":"sub-u","color":"amber","label":"differentiate","style":"solid"}]},"order":1,"title":"Define the substitution","description":"We choose $u = 2x + 3$ to simplify the exponent."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int e^{\\htmlData{anchor=orig-exp}{\\color{@sky}{2x+3}}} \\, \\htmlData{anchor=orig-dx}{\\color{@purple}{dx}}$"},{"latex":"$$= \\int e^{\\htmlData{anchor=new-exp}{\\color{@sky}{u}}} \\cdot \\htmlData{anchor=new-dx}{\\color{@purple}{\\frac{1}{2} \\, du}}$"},{"latex":"$$= \\frac{1}{2} \\int e^u \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"orig-exp"},{"color":"sky","style":"text-color","anchor":"new-exp"},{"color":"purple","style":"text-color","anchor":"orig-dx"},{"color":"purple","style":"text-color","anchor":"new-dx"}],"connections":[{"to":"new-exp","from":"orig-exp","color":"sky","label":"replace with u","style":"solid"},{"to":"new-dx","from":"orig-dx","color":"purple","label":"replace dx","style":"solid"}]},"order":2,"title":"Substitute into the integral","description":"We rewrite the entire integral in terms of our new variable, $u$."},{"type":"text","order":3,"title":"Perform the integration","content":"Now we integrate with respect to $u$. Recall that the integral of $e^u$ is simply $e^u$:\n\n$$\\frac{1}{2} \\int e^u \\, du = \\frac{1}{2} e^u + C$$\n\n*(M1)* is awarded for integrating the exponential function, and *(B1)* is awarded for including the constant of integration, $+ C$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{1}{2} e^{\\htmlData{anchor=u-term}{\\color{@sky}{u}}} + C$"},{"latex":"$$= \\frac{1}{2} e^{\\htmlData{anchor=x-term}{\\color{@sky}{2x + 3}}} + C$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-term"},{"color":"sky","style":"text-color","anchor":"x-term"}],"connections":[{"to":"x-term","from":"u-term","color":"sky","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to x","description":"Lastly, we substitute $u = 2x+3$ back into the expression to write the final answer in terms of $x$."},{"type":"text","order":5,"title":"State the final answer","content":"Our final integrated expression is:\n\n$$\\frac{1}{2}e^{2x+3} + C$$\n\n*(A1)* is awarded for finding the correct coefficient of $\\frac{1}{2}$."}]}],"generatedAt":"2026-06-09T05:57:48.342Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"4345a58e-3a7d-43ef-85bd-eec65a69839f","label":"b","content":"$$\\frac{1}{4x - 5}$","markscheme":"- Integrates to natural log: $\\frac{1}{4}\\ln|4x-5| + C$ 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":"Evidence of integrating to the form $k\\ln|4x-5|$.","code":"M"},{"id":"p2","marks":1,"description":"Correct coefficient of $\\frac{1}{4}$.","code":"A"},{"id":"p3","marks":1,"description":"Includes constant of integration $+ C$.","code":"B"}],"finalAnswers":["\\frac{1}{4}\\ln|4x-5| + C"],"generatedAt":"2026-05-06T14:04:13.116Z","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 expression with respect to $x$:\n\n$$\\int \\frac{1}{4x - 5} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = \\htmlData{anchor=u-expr}{\\color{@sky}{4x - 5}}$"},{"text":"Differentiating both sides with respect to $x$ gives:"},{"latex":"$$\\frac{du}{dx} = \\htmlData{anchor=deriv}{\\color{@amber}{4}}$"},{"text":"We can rearrange this to find an expression for $dx$:"},{"latex":"$$dx = \\htmlData{anchor=dx-expr}{\\color{@purple}{\\frac{1}{4} \\, du}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-expr"},{"color":"amber","style":"text-color","anchor":"deriv"},{"color":"purple","style":"text-color","anchor":"dx-expr"}],"connections":[{"to":"deriv","from":"u-expr","color":"amber","label":"derivative","style":"solid"},{"to":"dx-expr","from":"deriv","color":"purple","label":"solve for dx","style":"dashed"}]},"order":1,"title":"Define the substitution","description":"To simplify the integral, we introduce a new variable $u$ to represent the denominator."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int \\frac{1}{\\htmlData{anchor=sub-u}{\\color{@sky}{4x - 5}}} \\, \\htmlData{anchor=sub-dx}{\\color{@purple}{dx}}$"},{"text":"Substituting $u$ and $dx$ into our integral gives:"},{"latex":"$$\\int \\frac{1}{\\color{@sky}{u}} \\cdot \\color{@purple}{\\frac{1}{4} \\, du}$"},{"text":"We can pull the constant factor of $\\frac{1}{4}$ outside the integral:"},{"latex":"$$\\frac{1}{4} \\int \\frac{1}{u} \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-u"},{"color":"purple","style":"text-color","anchor":"sub-dx"}],"connections":[{"to":"sub-dx","from":"sub-u","color":"teal","label":null,"style":"solid"}]},"order":2,"title":"Substitute into the integral","description":"Now, we replace the original terms in $x$ with our new terms in $u$."},{"type":"text","order":3,"title":"Integrate with respect to u","content":"Using the standard integration rule $$\\int \\frac{1}{u} \\, du = \\ln|u| + C$$ we integrate our simplified expression:\n\n$$\\frac{1}{4} \\int \\frac{1}{u} \\, du = \\frac{1}{4}\\ln|u| + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{1}{4}\\ln|\\htmlData{anchor=final-u}{\\color{@sky}{u}}| + C$"},{"text":"Replacing $u$ with our original definition $u = 4x - 5$:"},{"latex":"$$\\frac{1}{4}\\ln|\\htmlData{anchor=final-ans}{\\color{@sky}{4x - 5}}| + C$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"final-u"},{"color":"sky","style":"text-color","anchor":"final-ans"}],"connections":[{"to":"final-ans","from":"final-u","color":"sky","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to x","description":"Finally, we substitute our original expression in $x$ back into the result."}]},{"id":"direct_integration","label":"Direct Integration","steps":[{"type":"text","order":0,"title":"Identify the standard rule","content":"We want to find the integral of the given expression:\n\n$$\\int \\frac{1}{4x - 5} \\, dx$$\n\nThis expression has a linear denominator of the form $ax + b$. We can directly integrate this using the standard logarithmic integration formula:\n\n$$\\int \\frac{1}{ax+b} \\, dx = \\frac{1}{a}\\ln|ax+b| + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\int \\frac{1}{\\htmlData{anchor=a-formula}{\\color{@sky}{a}}x + b} \\, dx = \\frac{1}{\\htmlData{anchor=a-coeff}{\\color{@sky}{a}}}\\ln|ax+b| + C$$"},{"latex":"$$$\\text{In our expression, the coefficient of } x \\text{ is } \\htmlData{anchor=a-val}{\\color{@sky}{4}}$$"},{"latex":"$$$\\int \\frac{1}{\\color{@sky}{4}x - 5} \\, dx = \\frac{1}{\\htmlData{anchor=a-result}{\\color{@sky}{4}}}\\ln|4x-5| + C$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-formula"},{"color":"sky","style":"text-color","anchor":"a-coeff"},{"color":"sky","style":"text-color","anchor":"a-val"},{"color":"sky","style":"text-color","anchor":"a-result"}],"connections":[{"to":"a-val","from":"a-formula","color":"sky","label":"identify a","style":"solid"},{"to":"a-result","from":"a-val","color":"sky","label":"substitute","style":"solid"}]},"order":1,"title":"Apply direct integration","description":"By matching the denominator $4x - 5$ to $ax + b$, we identify that $a = 4$. Substituting this into our integration formula yields the integrated logarithm term with its correct scale factor."},{"type":"text","order":2,"title":"State final answer and marks","content":"Let's review how marks are awarded for this problem to ensure we have met all the criteria:\n\n* **M1**: Integrating the expression to a natural logarithm form: $\\ln|4x-5|$\n* **A1**: Having the correct coefficient of $\\frac{1}{4}$ in front of the logarithm\n* **B1**: Writing down the constant of integration, $+ C$\n\nPutting everything together, we get our final answer:\n\n$$\\frac{1}{4}\\ln|4x-5| + C$$"}]}],"generatedAt":"2026-06-09T05:58:35.763Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"621eca7b-d33c-42b5-8d7e-104c9392b17b:reverse-chain-rule","label":"(a) Reverse Chain Rule","steps":[{"type":"text","order":0,"title":"Identify the standard rule","content":"To integrate an exponential function of the form $e^{ax+b}$, we can use the reverse chain rule.\n\nRecall that differentiating $e^{ax+b}$ with respect to $x$ yields $a e^{ax+b}$ by the chain rule. Therefore, when integrating, we do the reverse by dividing by the coefficient of $x$:\n\n$$\\int e^{ax+b} \\, dx = \\frac{1}{a} e^{ax+b} + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$e^{\\htmlData{anchor=a-val}{\\color{@sky}{2}}x + \\htmlData{anchor=b-val}{\\color{@amber}{3}}}$"},{"latex":"$$a = \\htmlData{anchor=a-sub}{\\color{@sky}{2}}, \\quad b = \\htmlData{anchor=b-sub}{\\color{@amber}{3}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-val"},{"color":"amber","style":"text-color","anchor":"b-val"}],"connections":[{"to":"a-sub","from":"a-val","color":"sky","label":null,"style":"solid"},{"to":"b-sub","from":"b-val","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Identify coefficients","description":"In our expression $e^{2x+3}$, we identify the coefficient of $x$ as $a = 2$ and the constant term as $b = 3$."},{"type":"text","order":2,"title":"Apply reverse chain rule","content":"Using our formula with $a = 2$:\n\n$$\\int e^{2x+3} \\, dx = \\frac{1}{2}e^{2x+3} + C$$\n\nThis step earns you the method mark (**M1**) for integrating the exponential and the accuracy mark (**A1**) for the correct coefficient of $\\frac{1}{2}$."},{"type":"text","order":3,"title":"Include integration constant","content":"Don't forget to include the constant of integration, $+ C$!\n\nThis final detail completes the answer and earns the remaining mark (**B1**):\n\n$$\\frac{1}{2}e^{2x+3} + C$$"}]},{"id":"621eca7b-d33c-42b5-8d7e-104c9392b17b:substitution","label":"(a) Integration by Substitution","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We want to integrate the given expression with respect to $x$:\n\n$$\\int e^{2x+3} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = \\htmlData{anchor=sub-u}{\\color{@sky}{2x + 3}}$"},{"text":"Now, we differentiate $u$ with respect to $x$ to find the relationship between $du$ and $dx$:"},{"latex":"$$\\frac{du}{dx} = \\htmlData{anchor=diff-u}{\\color{@amber}{2}} \\implies dx = \\htmlData{anchor=sub-dx}{\\color{@purple}{\\frac{1}{2}}} \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-u"},{"color":"purple","style":"text-color","anchor":"sub-dx"}],"connections":[{"to":"diff-u","from":"sub-u","color":"amber","label":"differentiate","style":"solid"}]},"order":1,"title":"Define the substitution","description":"We choose $u = 2x + 3$ to simplify the exponent."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int e^{\\htmlData{anchor=orig-exp}{\\color{@sky}{2x+3}}} \\, \\htmlData{anchor=orig-dx}{\\color{@purple}{dx}}$"},{"latex":"$$= \\int e^{\\htmlData{anchor=new-exp}{\\color{@sky}{u}}} \\cdot \\htmlData{anchor=new-dx}{\\color{@purple}{\\frac{1}{2} \\, du}}$"},{"latex":"$$= \\frac{1}{2} \\int e^u \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"orig-exp"},{"color":"sky","style":"text-color","anchor":"new-exp"},{"color":"purple","style":"text-color","anchor":"orig-dx"},{"color":"purple","style":"text-color","anchor":"new-dx"}],"connections":[{"to":"new-exp","from":"orig-exp","color":"sky","label":"replace with u","style":"solid"},{"to":"new-dx","from":"orig-dx","color":"purple","label":"replace dx","style":"solid"}]},"order":2,"title":"Substitute into the integral","description":"We rewrite the entire integral in terms of our new variable, $u$."},{"type":"text","order":3,"title":"Perform the integration","content":"Now we integrate with respect to $u$. Recall that the integral of $e^u$ is simply $e^u$:\n\n$$\\frac{1}{2} \\int e^u \\, du = \\frac{1}{2} e^u + C$$\n\n*(M1)* is awarded for integrating the exponential function, and *(B1)* is awarded for including the constant of integration, $+ C$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{1}{2} e^{\\htmlData{anchor=u-term}{\\color{@sky}{u}}} + C$"},{"latex":"$$= \\frac{1}{2} e^{\\htmlData{anchor=x-term}{\\color{@sky}{2x + 3}}} + C$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-term"},{"color":"sky","style":"text-color","anchor":"x-term"}],"connections":[{"to":"x-term","from":"u-term","color":"sky","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to x","description":"Lastly, we substitute $u = 2x+3$ back into the expression to write the final answer in terms of $x$."},{"type":"text","order":5,"title":"State the final answer","content":"Our final integrated expression is:\n\n$$\\frac{1}{2}e^{2x+3} + C$$\n\n*(A1)* is awarded for finding the correct coefficient of $\\frac{1}{2}$."}]},{"id":"4345a58e-3a7d-43ef-85bd-eec65a69839f:substitution","label":"(b) Integration by Substitution","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"We want to integrate the given expression with respect to $x$:\n\n$$\\int \\frac{1}{4x - 5} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = \\htmlData{anchor=u-expr}{\\color{@sky}{4x - 5}}$"},{"text":"Differentiating both sides with respect to $x$ gives:"},{"latex":"$$\\frac{du}{dx} = \\htmlData{anchor=deriv}{\\color{@amber}{4}}$"},{"text":"We can rearrange this to find an expression for $dx$:"},{"latex":"$$dx = \\htmlData{anchor=dx-expr}{\\color{@purple}{\\frac{1}{4} \\, du}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-expr"},{"color":"amber","style":"text-color","anchor":"deriv"},{"color":"purple","style":"text-color","anchor":"dx-expr"}],"connections":[{"to":"deriv","from":"u-expr","color":"amber","label":"derivative","style":"solid"},{"to":"dx-expr","from":"deriv","color":"purple","label":"solve for dx","style":"dashed"}]},"order":1,"title":"Define the substitution","description":"To simplify the integral, we introduce a new variable $u$ to represent the denominator."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int \\frac{1}{\\htmlData{anchor=sub-u}{\\color{@sky}{4x - 5}}} \\, \\htmlData{anchor=sub-dx}{\\color{@purple}{dx}}$"},{"text":"Substituting $u$ and $dx$ into our integral gives:"},{"latex":"$$\\int \\frac{1}{\\color{@sky}{u}} \\cdot \\color{@purple}{\\frac{1}{4} \\, du}$"},{"text":"We can pull the constant factor of $\\frac{1}{4}$ outside the integral:"},{"latex":"$$\\frac{1}{4} \\int \\frac{1}{u} \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-u"},{"color":"purple","style":"text-color","anchor":"sub-dx"}],"connections":[{"to":"sub-dx","from":"sub-u","color":"teal","label":null,"style":"solid"}]},"order":2,"title":"Substitute into the integral","description":"Now, we replace the original terms in $x$ with our new terms in $u$."},{"type":"text","order":3,"title":"Integrate with respect to u","content":"Using the standard integration rule $$\\int \\frac{1}{u} \\, du = \\ln|u| + C$$ we integrate our simplified expression:\n\n$$\\frac{1}{4} \\int \\frac{1}{u} \\, du = \\frac{1}{4}\\ln|u| + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{1}{4}\\ln|\\htmlData{anchor=final-u}{\\color{@sky}{u}}| + C$"},{"text":"Replacing $u$ with our original definition $u = 4x - 5$:"},{"latex":"$$\\frac{1}{4}\\ln|\\htmlData{anchor=final-ans}{\\color{@sky}{4x - 5}}| + C$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"final-u"},{"color":"sky","style":"text-color","anchor":"final-ans"}],"connections":[{"to":"final-ans","from":"final-u","color":"sky","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to x","description":"Finally, we substitute our original expression in $x$ back into the result."}]},{"id":"4345a58e-3a7d-43ef-85bd-eec65a69839f:direct_integration","label":"(b) Direct Integration","steps":[{"type":"text","order":0,"title":"Identify the standard rule","content":"We want to find the integral of the given expression:\n\n$$\\int \\frac{1}{4x - 5} \\, dx$$\n\nThis expression has a linear denominator of the form $ax + b$. We can directly integrate this using the standard logarithmic integration formula:\n\n$$\\int \\frac{1}{ax+b} \\, dx = \\frac{1}{a}\\ln|ax+b| + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\int \\frac{1}{\\htmlData{anchor=a-formula}{\\color{@sky}{a}}x + b} \\, dx = \\frac{1}{\\htmlData{anchor=a-coeff}{\\color{@sky}{a}}}\\ln|ax+b| + C$$"},{"latex":"$$$\\text{In our expression, the coefficient of } x \\text{ is } \\htmlData{anchor=a-val}{\\color{@sky}{4}}$$"},{"latex":"$$$\\int \\frac{1}{\\color{@sky}{4}x - 5} \\, dx = \\frac{1}{\\htmlData{anchor=a-result}{\\color{@sky}{4}}}\\ln|4x-5| + C$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-formula"},{"color":"sky","style":"text-color","anchor":"a-coeff"},{"color":"sky","style":"text-color","anchor":"a-val"},{"color":"sky","style":"text-color","anchor":"a-result"}],"connections":[{"to":"a-val","from":"a-formula","color":"sky","label":"identify a","style":"solid"},{"to":"a-result","from":"a-val","color":"sky","label":"substitute","style":"solid"}]},"order":1,"title":"Apply direct integration","description":"By matching the denominator $4x - 5$ to $ax + b$, we identify that $a = 4$. Substituting this into our integration formula yields the integrated logarithm term with its correct scale factor."},{"type":"text","order":2,"title":"State final answer and marks","content":"Let's review how marks are awarded for this problem to ensure we have met all the criteria:\n\n* **M1**: Integrating the expression to a natural logarithm form: $\\ln|4x-5|$\n* **A1**: Having the correct coefficient of $\\frac{1}{4}$ in front of the logarithm\n* **B1**: Writing down the constant of integration, $+ C$\n\nPutting everything together, we get our final answer:\n\n$$\\frac{1}{4}\\ln|4x-5| + C$$"}]}],"generatedAt":"2026-06-09T05:58:45.677Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":76},"walkthroughVideo":null},{"id":"7242d354-9652-4ce4-b93c-a86a27c4f362","revisionId":"b957f197-4c86-4b3b-96d2-9f36989e55e0","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"779b6cfe-e921-4545-a5d0-674afa30e173","label":"a","content":"Find: $\\int_0^2 \\frac{12}{(3x + 2)^2} dx$","markscheme":"- Rewrites as $12(3x+2)^{-2}$ and integrates: $\\left[ 12(3x+2)^{-1} \\times \\frac{1}{3} \\times (-1) \\right]_0^2 = \\left[ -4(3x+2)^{-1} \\right]_0^2$ M1 A1\n- Substitutes limits: $\\left[ -4(8)^{-1} \\right] - \\left[ -4(2)^{-1} \\right]$ M1\n- Calculates values: $-\\frac{1}{2} + 2$ M1\n- Final answer $\\frac{3}{2}$ or $1.5$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Evidence of rewriting the integrand as a power and attempting to integrate to the form $k(3x+2)^{-1}$.","code":"M"},{"id":"p2","marks":1,"description":"Correct integration including the factor of 1/3, resulting in $-4(3x+2)^{-1}$.","code":"A"},{"id":"p3","marks":1,"description":"Substitution of the limits 2 and 0 into their integrated expression.","code":"M"},{"id":"p4","marks":1,"description":"Correct evaluation of both limit terms before final simplification.","code":"M"},{"id":"p5","marks":1,"description":"Final simplified answer.","code":"A","acceptable":["3/2","1.5"]}],"finalAnswers":["3/2"],"generatedAt":"2026-05-06T14:04:10.493Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_integration","label":"Direct Integration (Reverse Chain Rule)","steps":[{"type":"text","order":0,"title":"Rewrite in index form","content":"Before integrating, it is helpful to rewrite the fraction in index form (with a negative exponent). \n\nWe rewrite the integrand $\\frac{12}{(3x+2)^2}$ as:\n\n$$12(3x+2)^{-2}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int 12(\\htmlData{anchor=linear-term}{\\color{@sky}{3x+2}})^{-2} dx$"},{"latex":"$$= \\frac{12}{-1}(\\color{@sky}{3x+2})^{-1} \\times \\htmlData{anchor=recip-deriv}{\\color{@amber}{\\frac{1}{3}}}$"},{"latex":"$$= \\htmlData{anchor=simplified-term}{\\color{@purple}{-4(3x+2)^{-1}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"linear-term"},{"color":"amber","style":"text-color","anchor":"recip-deriv"},{"color":"purple","style":"text-color","anchor":"simplified-term"}],"connections":[{"to":"recip-deriv","from":"linear-term","color":"amber","label":"divide by x coefficient","style":"dashed"}]},"order":1,"title":"Apply reverse chain rule","description":"To integrate $12(3x+2)^{-2}$ using the reverse chain rule, we raise the power of the bracket by 1 to get $-1$, divide by this new power, and also divide by the derivative of the inside bracket (which is $3$)."},{"type":"text","order":2,"title":"Set up definite integral","content":"Now we express the result using limits from $0$ to $2$. This earns the first method and accuracy marks ($M1$ and $A1$):\n\n$$\\left[ -4(3x+2)^{-1} \\right]_0^2 = \\left[ -\\frac{4}{3x+2} \\right]_0^2$$"},{"type":"text","order":3,"title":"Substitute the limits","content":"Next, substitute the upper limit $x = 2$ and the lower limit $x = 0$ into the expression ($M1$):\n\n$$\\left( -\\frac{4}{3(2)+2} \\right) - \\left( -\\frac{4}{3(0)+2} \\right)$$\n\nSimplify the terms inside the brackets:\n\n$$\\left( -\\frac{4}{8} \\right) - \\left( -\\frac{4}{2} \\right)$$"},{"type":"text","order":4,"title":"Calculate final value","content":"Simplify the fractions and calculate the final result ($M1$):\n\n$$-\\frac{1}{2} - (-2) = -\\frac{1}{2} + 2 = \\frac{3}{2}$$\n\nThus, the final evaluation of the definite integral is $\\frac{3}{2}$ or $1.5$ ($A1$).","calculator":[{"gdc":{"expression":"\\int(12/(3x+2)^2, 0, 2)","instructions":"Go to the Run-Matrix menu, press [Option], then [Calculus] (F4), then [d/dx] (F4) or [Integral] (F4 depending on OS). Enter the integral expression with lower limit 0 and upper limit 2."},"ti84":{"expression":"fnInt(12/(3X+2)^2, X, 0, 2)","instructions":"Press [MATH] and select 9:fnInt(. Enter the function, variable, and limits as fnInt(12/(3X+2)^2, X, 0, 2) and press [ENTER]."},"desmos":{"expression":"\\int_{0}^{2}\\frac{12}{\\left(3x+2\\right)^{2}}dx","instructions":"Type 'int' to generate the integral symbol, input 0 as the lower limit and 2 as the upper limit, then type the integrand: 12/(3x+2)^2 dx."},"description":"Evaluate the definite integral using a graphing calculator."}]}]},{"id":"u_substitution","label":"Integration by u-Substitution","steps":[{"type":"text","order":0,"title":"Define the substitution","content":"To solve this integral using substitution, we choose a new variable, $u$, to simplify the integrand.\n\nLet:\n$$u = 3x + 2$$\n\nNext, we find the derivative of $u$ with respect to $x$ to express $dx$ in terms of $du$:\n$$\\frac{du}{dx} = 3 \\implies dx = \\frac{1}{3} du$$"},{"type":"text","order":1,"title":"Transform the limits","content":"Since we are changing the variable from $x$ to $u$, we must also transform the limits of integration ($x = 0$ and $x = 2$) using our substitution formula $u = 3x + 2$:\n\n* **Lower limit:** When $x = 0$:\n $$u = 3(0) + 2 = 2$$\n* **Upper limit:** When $x = 2$:\n $$u = 3(2) + 2 = 8$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\int_{\\htmlData{anchor=low-x}{0}}^{\\htmlData{anchor=up-x}{2}} \\frac{12}{(\\htmlData{anchor=sub-x}{3x + 2})^2} \\, \\htmlData{anchor=dx-orig}{dx}$$"},{"latex":"$$$\\int_{\\htmlData{anchor=low-u}{2}}^{\\htmlData{anchor=up-u}{8}} \\frac{12}{\\htmlData{anchor=sub-u}{u}^2} \\left( \\htmlData{anchor=dx-new}{\\frac{1}{3} du} \\right)$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"low-u"},{"color":"pink","style":"text-color","anchor":"up-u"},{"color":"amber","style":"text-color","anchor":"sub-u"},{"color":"purple","style":"text-color","anchor":"dx-new"}],"connections":[{"to":"low-u","from":"low-x","color":"sky","label":"lower limit","style":"solid"},{"to":"up-u","from":"up-x","color":"pink","label":"upper limit","style":"solid"},{"to":"sub-u","from":"sub-x","color":"amber","label":"u = 3x+2","style":"solid"},{"to":"dx-new","from":"dx-orig","color":"purple","label":"dx = 1/3 du","style":"solid"}]},"order":2,"title":"Rewrite the integral","description":"Substitute $u$, $dx$, and the new limits of integration into the original integral expression."},{"type":"text","order":3,"title":"Simplify the integrand","content":"Let's simplify the constant factors in our new integral expression:\n\n$$\\int_{2}^{8} \\frac{12}{u^2} \\cdot \\frac{1}{3} \\, du = \\int_{2}^{8} 4u^{-2} \\, du$$"},{"type":"text","order":4,"title":"Integrate with respect to u","content":"Now we integrate $4u^{-2}$ with respect to $u$ by applying the power rule of integration, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1}$:\n\n$$\\int 4u^{-2} \\, du = \\left[ \\frac{4u^{-1}}{-1} \\right] = \\left[ -4u^{-1} \\right]$$\n\nApplying this to our definite integral limits from $2$ to $8$:\n$$\\left[ -\\frac{4}{u} \\right]_{2}^{8}$$"},{"type":"text","order":5,"title":"Evaluate the limits","content":"Substitute the upper limit $8$ and lower limit $2$ into the integrated expression:\n\n$$\\left( -\\frac{4}{8} \\right) - \\left( -\\frac{4}{2} \\right)$$\n\nSimplify each term:\n$$= -\\frac{1}{2} - (-2)$$\n$$= -\\frac{1}{2} + 2 = \\frac{3}{2}$$\n\nSo, the final value of the definite integral is **$\\frac{3}{2}$** (or **$1.5$**).","calculator":[{"gdc":{"expression":"∫(12/(3x+2)², 0, 2)","instructions":"Go to the Run-Matrix menu, press [OPTN], [F4] (CALC), [F4] (∫dx). Enter the function 12/(3x+2)², set the lower bound as 0, and the upper bound as 2."},"ti84":{"expression":"fnInt(12/(3X+2)², X, 0, 2)","instructions":"Press [MATH], select 9:fnInt(. Input the parameters as fnInt(12/(3X+2)², X, 0, 2) and press [ENTER]."},"desmos":{"expression":"\\int_{0}^{2}\\frac{12}{\\left(3x+2\\right)^{2}}dx","instructions":"Type 'int' to create an integral symbol. Put 0 in the lower limit box and 2 in the upper limit box. Inside the integral, type '12/(3x+2)^2 dx'."},"description":"Verify the value of the definite integral using a graphing calculator."}]}]}],"generatedAt":"2026-06-09T06:04:26.203Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"779b6cfe-e921-4545-a5d0-674afa30e173:direct_integration","label":"Direct Integration (Reverse Chain Rule)","steps":[{"type":"text","order":0,"title":"Rewrite in index form","content":"Before integrating, it is helpful to rewrite the fraction in index form (with a negative exponent). \n\nWe rewrite the integrand $\\frac{12}{(3x+2)^2}$ as:\n\n$$12(3x+2)^{-2}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int 12(\\htmlData{anchor=linear-term}{\\color{@sky}{3x+2}})^{-2} dx$"},{"latex":"$$= \\frac{12}{-1}(\\color{@sky}{3x+2})^{-1} \\times \\htmlData{anchor=recip-deriv}{\\color{@amber}{\\frac{1}{3}}}$"},{"latex":"$$= \\htmlData{anchor=simplified-term}{\\color{@purple}{-4(3x+2)^{-1}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"linear-term"},{"color":"amber","style":"text-color","anchor":"recip-deriv"},{"color":"purple","style":"text-color","anchor":"simplified-term"}],"connections":[{"to":"recip-deriv","from":"linear-term","color":"amber","label":"divide by x coefficient","style":"dashed"}]},"order":1,"title":"Apply reverse chain rule","description":"To integrate $12(3x+2)^{-2}$ using the reverse chain rule, we raise the power of the bracket by 1 to get $-1$, divide by this new power, and also divide by the derivative of the inside bracket (which is $3$)."},{"type":"text","order":2,"title":"Set up definite integral","content":"Now we express the result using limits from $0$ to $2$. This earns the first method and accuracy marks ($M1$ and $A1$):\n\n$$\\left[ -4(3x+2)^{-1} \\right]_0^2 = \\left[ -\\frac{4}{3x+2} \\right]_0^2$$"},{"type":"text","order":3,"title":"Substitute the limits","content":"Next, substitute the upper limit $x = 2$ and the lower limit $x = 0$ into the expression ($M1$):\n\n$$\\left( -\\frac{4}{3(2)+2} \\right) - \\left( -\\frac{4}{3(0)+2} \\right)$$\n\nSimplify the terms inside the brackets:\n\n$$\\left( -\\frac{4}{8} \\right) - \\left( -\\frac{4}{2} \\right)$$"},{"type":"text","order":4,"title":"Calculate final value","content":"Simplify the fractions and calculate the final result ($M1$):\n\n$$-\\frac{1}{2} - (-2) = -\\frac{1}{2} + 2 = \\frac{3}{2}$$\n\nThus, the final evaluation of the definite integral is $\\frac{3}{2}$ or $1.5$ ($A1$).","calculator":[{"gdc":{"expression":"\\int(12/(3x+2)^2, 0, 2)","instructions":"Go to the Run-Matrix menu, press [Option], then [Calculus] (F4), then [d/dx] (F4) or [Integral] (F4 depending on OS). Enter the integral expression with lower limit 0 and upper limit 2."},"ti84":{"expression":"fnInt(12/(3X+2)^2, X, 0, 2)","instructions":"Press [MATH] and select 9:fnInt(. Enter the function, variable, and limits as fnInt(12/(3X+2)^2, X, 0, 2) and press [ENTER]."},"desmos":{"expression":"\\int_{0}^{2}\\frac{12}{\\left(3x+2\\right)^{2}}dx","instructions":"Type 'int' to generate the integral symbol, input 0 as the lower limit and 2 as the upper limit, then type the integrand: 12/(3x+2)^2 dx."},"description":"Evaluate the definite integral using a graphing calculator."}]}]},{"id":"779b6cfe-e921-4545-a5d0-674afa30e173:u_substitution","label":"Integration by u-Substitution","steps":[{"type":"text","order":0,"title":"Define the substitution","content":"To solve this integral using substitution, we choose a new variable, $u$, to simplify the integrand.\n\nLet:\n$$u = 3x + 2$$\n\nNext, we find the derivative of $u$ with respect to $x$ to express $dx$ in terms of $du$:\n$$\\frac{du}{dx} = 3 \\implies dx = \\frac{1}{3} du$$"},{"type":"text","order":1,"title":"Transform the limits","content":"Since we are changing the variable from $x$ to $u$, we must also transform the limits of integration ($x = 0$ and $x = 2$) using our substitution formula $u = 3x + 2$:\n\n* **Lower limit:** When $x = 0$:\n $$u = 3(0) + 2 = 2$$\n* **Upper limit:** When $x = 2$:\n $$u = 3(2) + 2 = 8$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\int_{\\htmlData{anchor=low-x}{0}}^{\\htmlData{anchor=up-x}{2}} \\frac{12}{(\\htmlData{anchor=sub-x}{3x + 2})^2} \\, \\htmlData{anchor=dx-orig}{dx}$$"},{"latex":"$$$\\int_{\\htmlData{anchor=low-u}{2}}^{\\htmlData{anchor=up-u}{8}} \\frac{12}{\\htmlData{anchor=sub-u}{u}^2} \\left( \\htmlData{anchor=dx-new}{\\frac{1}{3} du} \\right)$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"low-u"},{"color":"pink","style":"text-color","anchor":"up-u"},{"color":"amber","style":"text-color","anchor":"sub-u"},{"color":"purple","style":"text-color","anchor":"dx-new"}],"connections":[{"to":"low-u","from":"low-x","color":"sky","label":"lower limit","style":"solid"},{"to":"up-u","from":"up-x","color":"pink","label":"upper limit","style":"solid"},{"to":"sub-u","from":"sub-x","color":"amber","label":"u = 3x+2","style":"solid"},{"to":"dx-new","from":"dx-orig","color":"purple","label":"dx = 1/3 du","style":"solid"}]},"order":2,"title":"Rewrite the integral","description":"Substitute $u$, $dx$, and the new limits of integration into the original integral expression."},{"type":"text","order":3,"title":"Simplify the integrand","content":"Let's simplify the constant factors in our new integral expression:\n\n$$\\int_{2}^{8} \\frac{12}{u^2} \\cdot \\frac{1}{3} \\, du = \\int_{2}^{8} 4u^{-2} \\, du$$"},{"type":"text","order":4,"title":"Integrate with respect to u","content":"Now we integrate $4u^{-2}$ with respect to $u$ by applying the power rule of integration, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1}$:\n\n$$\\int 4u^{-2} \\, du = \\left[ \\frac{4u^{-1}}{-1} \\right] = \\left[ -4u^{-1} \\right]$$\n\nApplying this to our definite integral limits from $2$ to $8$:\n$$\\left[ -\\frac{4}{u} \\right]_{2}^{8}$$"},{"type":"text","order":5,"title":"Evaluate the limits","content":"Substitute the upper limit $8$ and lower limit $2$ into the integrated expression:\n\n$$\\left( -\\frac{4}{8} \\right) - \\left( -\\frac{4}{2} \\right)$$\n\nSimplify each term:\n$$= -\\frac{1}{2} - (-2)$$\n$$= -\\frac{1}{2} + 2 = \\frac{3}{2}$$\n\nSo, the final value of the definite integral is **$\\frac{3}{2}$** (or **$1.5$**).","calculator":[{"gdc":{"expression":"∫(12/(3x+2)², 0, 2)","instructions":"Go to the Run-Matrix menu, press [OPTN], [F4] (CALC), [F4] (∫dx). Enter the function 12/(3x+2)², set the lower bound as 0, and the upper bound as 2."},"ti84":{"expression":"fnInt(12/(3X+2)², X, 0, 2)","instructions":"Press [MATH], select 9:fnInt(. Input the parameters as fnInt(12/(3X+2)², X, 0, 2) and press [ENTER]."},"desmos":{"expression":"\\int_{0}^{2}\\frac{12}{\\left(3x+2\\right)^{2}}dx","instructions":"Type 'int' to create an integral symbol. Put 0 in the lower limit box and 2 in the upper limit box. Inside the integral, type '12/(3x+2)^2 dx'."},"description":"Verify the value of the definite integral using a graphing calculator."}]}]}],"generatedAt":"2026-06-09T06:04:39.659Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":35},"walkthroughVideo":null}],"subjectCode":"a_level_maths__ccea","summaries":[{"id":"41fb9ee5-982a-4e7b-ba09-a06e0a62b120","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"1ad2598d-4168-4c2d-9266-61dbe77513b6","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"66d1554b-b243-47c0-a69a-9d7efc1f3c97","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"678b8a53-192a-4dc8-8642-f373e1ea4601","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6a3c5f33-7ee6-4203-9e71-d6f699fe93be","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"7242d354-9652-4ce4-b93c-a86a27c4f362","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"94974db3-2c8e-482a-9290-65838478f475","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":false},{"id":"9d98ac71-67b3-4622-a4f2-a035d1537bdf","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"c4e6b947-384c-4f92-8ef3-0ef73e7dafb0","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"e4aae231-ae0a-4dde-89b7-edf0e38302b4","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e590d6fd-a354-4b0f-b68f-e586836db738","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"e7766021-317c-4299-8e8f-b50a9ec0b559","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1c062306-1388-4be0-af8a-f4aee2770406","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"71396e06-cadf-4d46-b502-86a4b22086a2","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"3b1cdc8b-dd67-45bf-9f55-ab1c7970a359","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"3eacbbf9-f07a-4d1f-8856-766abf8a4f98","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"564330e4-3146-4624-92ba-7e0a988422c0","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"82d47be0-9de4-468e-b752-a583b4bc530e","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"946e6230-7a3d-4000-b391-7ce2b0b2fff0","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"bdc8d91d-c276-4450-9b9c-f694f8a23cf2","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d929e877-ba83-4854-ad2c-efd7438df34b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"e0b0e605-1cb8-406c-a66b-dd1dcd10a666","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"e2a4f4b3-0e2e-4895-ba94-4348ce074c2c","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"e8e07c8c-2869-47ae-b49d-30874035a15e","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":false},{"id":"032b71c5-1f39-4b32-8302-1b7b7935fe15","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"1000ed6d-fe65-4492-9d9e-726a40b78c5a","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"16baf928-c5e9-4af9-85de-ebb2b2243183","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"21b2b410-e76f-498c-bc49-84474bad9107","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2af416db-89f5-4bef-bcc6-a4891fd8c4cf","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"343dd622-7848-4031-bd4e-6d6e5e38de8c","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"3c77626d-a364-4fa7-8697-8aa18fceb88c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"63ea12ec-34dc-47ab-9af8-68de7fc877e8","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":fa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Integration (A-level only)","topicId":"53edcf23-243a-46a3-8a7b-03d789a50f35","topicSidebar":"$L3c","topicSlug":"integration-6","topicTitle":"3.6.1 Integration (A-level only)"}]}],"$L3d","$L3e"]}]

3.6.1 Integration (A-level only)

Question 1

3.6.1 Integration (A-level only) Questions

3.6.1 Integration (A-level only)

Question 1

3.6.1 Integration (A-level only) Questions