to $u$ using the power rule for integration:\n\n$$\\int u^3 du = \\frac{1}{4}u^4 + C$$\n\nwhere $C$ is the constant of integration."},{"type":"text","order":3,"title":"Substitute back for x","content":"Finally, we substitute our original expression for $u$ back into the result to obtain the final answer in terms of $x$:\n\n$$\\frac{1}{4}(e^x + 2)^4 + C$$"}]},{"id":"44629351-4ff8-40a9-bcfc-c0907285c59b:dx_rearrangement","label":"Substitution by Expressing $dx$","steps":[{"type":"text","order":0,"title":"Define the substitution","content":"We are given the substitution $u = e^x + 2$. First, we find the derivative of $u$ with respect to $x$:\n\n$$\\frac{du}{dx} = e^x$$\n\nThis step earns the first mark **(M1)** for correctly differentiating the substitution.","highlights":[{"text":"u = e^x + 2","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{du}{dx} = \\htmlData{anchor=diff-exp}{\\color{@sky}{e^x}}$"},{"latex":"$$dx = \\frac{du}{\\htmlData{anchor=dx-exp}{\\color{@sky}{e^x}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"diff-exp"},{"color":"sky","style":"text-color","anchor":"dx-exp"}],"connections":[{"to":"dx-exp","from":"diff-exp","color":"sky","label":"rearrange","style":"solid"}]},"order":1,"title":"Express dx in terms of du","description":"Next, we rearrange the derivative equation to express $dx$ in terms of $du$ and $e^x$."},{"type":"text","order":2,"title":"Substitute into the integral","content":"Now, we substitute $u = e^x + 2$ and $dx = \\frac{du}{e^x}$ into the original integral:\n\n$$\\int e^x (e^x + 2)^3 dx = \\int e^x \\cdot u^3 \\cdot \\frac{du}{e^x}$$\n\nNotice how the $e^x$ terms cancel out beautifully:\n\n$$\\int \\cancel{e^x} \\cdot u^3 \\cdot \\frac{du}{\\cancel{e^x}} = \\int u^3 du$$\n\nThis cancellation simplifies our integral to being purely in terms of $u$. This step earns the second mark **(M1)**."},{"type":"text","order":3,"title":"Integrate with respect to u","content":"Using the power rule for integration, we integrate $u^3$:\n\n$$\\int u^3 du = \\frac{1}{4}u^4 + C$$\n\nwhere $C$ is the constant of integration. This integration step earns another mark **(M1)**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Since } \\htmlData{anchor=u-defn}{\\color{@purple}{u}} = \\htmlData{anchor=u-val}{\\color{@purple}{e^x + 2}}$"},{"latex":"$$\\frac{1}{4}\\htmlData{anchor=u-sub}{\\color{@purple}{u}}^4 + C = \\frac{1}{4}(\\htmlData{anchor=u-final}{\\color{@purple}{e^x + 2}})^4 + C$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"u-defn"},{"color":"purple","style":"text-color","anchor":"u-sub"}],"connections":[{"to":"u-final","from":"u-val","color":"purple","label":"substitute back","style":"solid"}]},"order":4,"title":"Substitute back to find the final answer","description":"To complete the problem, replace $u$ with the original expression in terms of $x$ to get the final answer. This earns the final two marks **(A2)**."}]}],"generatedAt":"2026-06-11T03:09:58.902Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"d2d74518-ad96-4bb8-931a-2b1e293a3b43","revisionId":"694e8816-99b1-4036-b14e-84bf37de442c","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":6,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"2560fdd0-80db-408e-af70-cf59138a2001","label":"a","content":"Find the general solution to the differential equation $\\frac{dy}{dx} = xy e^{-x}$","markscheme":"- Separates variables: $\\int \\frac{1}{y} dy = \\int x e^{-x} dx$ M1\n- Integrates LHS: $\\ln y$ (or $\\ln|y|$) B1\n- Integrates RHS by parts: $u=x, v'=e^{-x} \\implies u'=1, v=-e^{-x}$ M1\n- RHS $= -x e^{-x} - \\int -e^{-x} dx = -x e^{-x} - e^{-x} + c$ A2\n- Final solution: $\\ln y = -e^{-x}(x + 1) + c$ A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Separation of variables.","code":"M"},{"id":"p2","marks":1,"descriptio

Integration

Question 79

Integration Questions

Integration

Question 79

Integration Questions