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bg-background","children":["$","$L3b",null,{"backHref":"/a-level/maths/ccea/questions","basePath":"/a-level/maths/ccea/integration-9/questions","formulaUrl":"https://ccea.org.uk/downloads/docs/Support/General/2019/Mathematical%20Formulae%20and%20Tables.pdf","initialIndex":20,"initialQuestionWindow":[{"id":"cf8a0e30-159c-4ad3-aeee-9692e1ae78ae","revisionId":"35779ecb-815b-47e6-bb77-ea9c6fe47f49","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"9dd1e702-ba8d-446c-a1a7-74400e7b94e7","label":"a","content":"Use the substitution $u = x^2 + 5$ to find $\\int 2x(x^2 + 5)^4 dx$","markscheme":"$$u = x^2 + 5 \\implies \\frac{du}{dx} = 2x$ M1\n$\\int 2x(u)^4 \\frac{1}{2x} du = \\int u^4 du$ M1\n$= \\frac{1}{5}u^5 + c$ M1\n$= \\frac{1}{5}(x^2 + 5)^5 + c$ A2","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Differentiates the substitution $u = x^2 + 5$ to find $\\frac{du}{dx}$","code":"M"},{"id":"p2","marks":1,"description":"Correctly substitutes $u$ and $dx$ into the integral to eliminate $x$","code":"M"},{"id":"p3","marks":1,"description":"Integrates the resulting power of $u$","code":"M"},{"id":"p4","marks":2,"description":"Replaces $u$ with the original expression in $x$ and includes the constant of integration","code":"A"}],"finalAnswers":["\\frac{1}{5}(x^2 + 5)^5 + c"],"generatedAt":"2026-05-06T14:04:27.121Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"differential_substitution","label":"Direct Differential Substitution","steps":[{"type":"text","order":0,"title":"Identify substitution and differential","content":"The question asks us to use the substitution $u = x^2 + 5$.\n\nFirst, let's find the derivative of $u$ with respect to $x$:\n\n$$\\frac{du}{dx} = 2x$$\n\nIn differential form, this can be written directly as:\n\n$$du = 2x\\,dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int \\htmlData{anchor=diff-comp}{\\color{@sky}{2x}} \\left(\\htmlData{anchor=u-inner}{\\color{@amber}{x^2 + 5}}\\right)^4 \\htmlData{anchor=dx-comp}{\\color{@sky}{dx}}$"},{"latex":"$$\\int \\left(\\htmlData{anchor=u-sub}{\\color{@amber}{u}}\\right)^4 \\htmlData{anchor=du-sub}{\\color{@sky}{du}}$"}],"highlights":[{"color":"amber","style":"box","anchor":"u-inner"},{"color":"sky","style":"box","anchor":"diff-comp"}],"connections":[{"to":"u-sub","from":"u-inner","color":"amber","label":"substitute u","style":"solid"},{"to":"du-sub","from":"diff-comp","color":"sky","label":"substitute du","style":"solid"}]},"order":1,"title":"Substitute into the integral","description":"We rewrite the original integral to group the differential terms together:\n\n$$\\int (x^2 + 5)^4 \\cdot (2x\\,dx)$$\n\nNow, we substitute $u = x^2 + 5$ and the differential component $du = 2x\\,dx$ directly into the integral."},{"type":"text","order":2,"title":"Integrate with respect to u","content":"Now we integrate $u^4$ with respect to $u$ using the standard power rule for integration, $\\int u^n du = \\frac{u^{n+1}}{n+1} + c$:\n\n$$\\int u^4 du = \\frac{1}{5}u^5 + c$$\n\nwhere $c$ is the constant of integration."},{"type":"text","order":3,"title":"Substitute back to x","content":"Finally, replace $u$ with our original expression in terms of $x$, which is $u = x^2 + 5$:\n\n$$\\frac{1}{5}(x^2 + 5)^5 + c$$"}]},{"id":"dx_substitution","label":"Explicit dx Substitution","steps":[{"type":"text","order":0,"title":"Differentiate the substitution","content":"We are given the substitution $u = x^2 + 5$. First, differentiate $u$ with respect to $x$ to find $\\frac{du}{dx}$:\n\n$$\\frac{du}{dx} = 2x$$\n\nThis step helps us connect the differentials $dx$ and $du$, earning you the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{\\htmlData{anchor=du-term}{du}}{\\htmlData{anchor=dx-term}{\\color{@sky}{dx}}} = \\htmlData{anchor=deriv-term}{\\color{@amber}{2x}}$"},{"latex":"$$\\htmlData{anchor=dx-rearranged}{\\color{@sky}{dx}} = \\frac{1}{\\htmlData{anchor=deriv-rearranged}{\\color{@amber}{2x}}}\\,\\htmlData{anchor=du-rearranged}{du}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"dx-term"},{"color":"amber","style":"text-color","anchor":"deriv-term"}],"connections":[{"to":"dx-rearranged","from":"dx-term","color":"sky","label":"rearrange","style":"solid"},{"to":"deriv-rearranged","from":"deriv-term","color":"amber","label":"divide","style":"solid"}]},"order":1,"title":"Rearrange to express dx","description":"Now, rearrange the derivative equation to explicitly express $dx$ in terms of $du$ and $x$."},{"type":"text","order":2,"title":"Substitute into the integral","content":"Now we substitute both $u = x^2 + 5$ and $dx = \\frac{1}{2x}\\,du$ into our original integral:\n\n$$\\int 2x(x^2 + 5)^4 \\, dx = \\int 2x \\cdot (u)^4 \\cdot \\left(\\frac{1}{2x}\\,du\\right)$$\n\nNotice that the $2x$ term in the numerator cancels perfectly with the $2x$ in the denominator:\n\n$$\\int \\cancel{2x} \\cdot u^4 \\cdot \\frac{1}{\\cancel{2x}}\\,du = \\int u^4 \\, du$$\n\nThis algebraic simplification earns the second method mark."},{"type":"text","order":3,"title":"Integrate with respect to u","content":"Next, integrate $u^4$ with respect to $u$ using the power rule for integration, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1} + c$:\n\n$$\\int u^4 \\, du = \\frac{1}{5}u^5 + c$$\n\n*Remember to always include the constant of integration, $+ c$.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=u-var}{\\color{@purple}{u}} = \\htmlData{anchor=x-expr}{\\color{@purple}{x^2 + 5}}$"},{"latex":"$$\\frac{1}{5}(\\htmlData{anchor=u-sub}{\\color{@purple}{u}})^5 + c = \\frac{1}{5}(\\htmlData{anchor=x-sub}{\\color{@purple}{x^2 + 5}})^5 + c$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"u-var"},{"color":"purple","style":"text-color","anchor":"x-expr"}],"connections":[{"to":"x-sub","from":"x-expr","color":"purple","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to x","description":"Finally, substitute the original expression for $u$ back into our integrated result to get the final answer in terms of $x$."}]}],"generatedAt":"2026-06-09T02:00:26.438Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"9dd1e702-ba8d-446c-a1a7-74400e7b94e7:differential_substitution","label":"Direct Differential Substitution","steps":[{"type":"text","order":0,"title":"Identify substitution and differential","content":"The question asks us to use the substitution $u = x^2 + 5$.\n\nFirst, let's find the derivative of $u$ with respect to $x$:\n\n$$\\frac{du}{dx} = 2x$$\n\nIn differential form, this can be written directly as:\n\n$$du = 2x\\,dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int \\htmlData{anchor=diff-comp}{\\color{@sky}{2x}} \\left(\\htmlData{anchor=u-inner}{\\color{@amber}{x^2 + 5}}\\right)^4 \\htmlData{anchor=dx-comp}{\\color{@sky}{dx}}$"},{"latex":"$$\\int \\left(\\htmlData{anchor=u-sub}{\\color{@amber}{u}}\\right)^4 \\htmlData{anchor=du-sub}{\\color{@sky}{du}}$"}],"highlights":[{"color":"amber","style":"box","anchor":"u-inner"},{"color":"sky","style":"box","anchor":"diff-comp"}],"connections":[{"to":"u-sub","from":"u-inner","color":"amber","label":"substitute u","style":"solid"},{"to":"du-sub","from":"diff-comp","color":"sky","label":"substitute du","style":"solid"}]},"order":1,"title":"Substitute into the integral","description":"We rewrite the original integral to group the differential terms together:\n\n$$\\int (x^2 + 5)^4 \\cdot (2x\\,dx)$$\n\nNow, we substitute $u = x^2 + 5$ and the differential component $du = 2x\\,dx$ directly into the integral."},{"type":"text","order":2,"title":"Integrate with respect to u","content":"Now we integrate $u^4$ with respect to $u$ using the standard power rule for integration, $\\int u^n du = \\frac{u^{n+1}}{n+1} + c$:\n\n$$\\int u^4 du = \\frac{1}{5}u^5 + c$$\n\nwhere $c$ is the constant of integration."},{"type":"text","order":3,"title":"Substitute back to x","content":"Finally, replace $u$ with our original expression in terms of $x$, which is $u = x^2 + 5$:\n\n$$\\frac{1}{5}(x^2 + 5)^5 + c$$"}]},{"id":"9dd1e702-ba8d-446c-a1a7-74400e7b94e7:dx_substitution","label":"Explicit dx Substitution","steps":[{"type":"text","order":0,"title":"Differentiate the substitution","content":"We are given the substitution $u = x^2 + 5$. First, differentiate $u$ with respect to $x$ to find $\\frac{du}{dx}$:\n\n$$\\frac{du}{dx} = 2x$$\n\nThis step helps us connect the differentials $dx$ and $du$, earning you the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{\\htmlData{anchor=du-term}{du}}{\\htmlData{anchor=dx-term}{\\color{@sky}{dx}}} = \\htmlData{anchor=deriv-term}{\\color{@amber}{2x}}$"},{"latex":"$$\\htmlData{anchor=dx-rearranged}{\\color{@sky}{dx}} = \\frac{1}{\\htmlData{anchor=deriv-rearranged}{\\color{@amber}{2x}}}\\,\\htmlData{anchor=du-rearranged}{du}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"dx-term"},{"color":"amber","style":"text-color","anchor":"deriv-term"}],"connections":[{"to":"dx-rearranged","from":"dx-term","color":"sky","label":"rearrange","style":"solid"},{"to":"deriv-rearranged","from":"deriv-term","color":"amber","label":"divide","style":"solid"}]},"order":1,"title":"Rearrange to express dx","description":"Now, rearrange the derivative equation to explicitly express $dx$ in terms of $du$ and $x$."},{"type":"text","order":2,"title":"Substitute into the integral","content":"Now we substitute both $u = x^2 + 5$ and $dx = \\frac{1}{2x}\\,du$ into our original integral:\n\n$$\\int 2x(x^2 + 5)^4 \\, dx = \\int 2x \\cdot (u)^4 \\cdot \\left(\\frac{1}{2x}\\,du\\right)$$\n\nNotice that the $2x$ term in the numerator cancels perfectly with the $2x$ in the denominator:\n\n$$\\int \\cancel{2x} \\cdot u^4 \\cdot \\frac{1}{\\cancel{2x}}\\,du = \\int u^4 \\, du$$\n\nThis algebraic simplification earns the second method mark."},{"type":"text","order":3,"title":"Integrate with respect to u","content":"Next, integrate $u^4$ with respect to $u$ using the power rule for integration, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1} + c$:\n\n$$\\int u^4 \\, du = \\frac{1}{5}u^5 + c$$\n\n*Remember to always include the constant of integration, $+ c$.*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=u-var}{\\color{@purple}{u}} = \\htmlData{anchor=x-expr}{\\color{@purple}{x^2 + 5}}$"},{"latex":"$$\\frac{1}{5}(\\htmlData{anchor=u-sub}{\\color{@purple}{u}})^5 + c = \\frac{1}{5}(\\htmlData{anchor=x-sub}{\\color{@purple}{x^2 + 5}})^5 + c$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"u-var"},{"color":"purple","style":"text-color","anchor":"x-expr"}],"connections":[{"to":"x-sub","from":"x-expr","color":"purple","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to x","description":"Finally, substitute the original expression for $u$ back into our integrated result to get the final answer in terms of $x$."}]}],"generatedAt":"2026-06-09T02:00:49.640Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"0f13d61c-9b51-4ddd-8a76-5b68c239d9fd","revisionId":"af868bc5-39dc-467c-a11a-f95f6eae4334","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"64366fa3-eba8-4c8b-a669-6b9bea4eeeb3","label":"a","content":"Use the substitution $u = x^3 + 1$ to find $\\int 3x^2(x^3 + 1)^3 dx$","markscheme":"$$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 + 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