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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":30,"initialQuestionWindow":[{"id":"ba27ece0-83bb-41ef-99b4-f11f5c223533","revisionId":"bf73c2ee-612b-4bcc-823d-a8e97f37fd11","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"9974450d-b64e-4b3e-8e35-11698c71cdff","label":"a","content":"Use integration by parts to find $\\int x e^x \\, dx$.","markscheme":"- Sets $u = x$ and $\\frac{dv}{dx} = e^x$, so $\\frac{du}{dx} = 1$ and $v = e^x$ M1\n- Applies the integration by parts formula: $x e^x - \\int e^x \\, dx$ M1\n- Integrates $e^x$ to get $e^x$ A1\n- Final answer: $x e^x - e^x + c$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Identifies $u = x$ and $\\frac{dv}{dx} = e^x$ and finds correct derivatives/integrals.","code":"M"},{"id":"p2","marks":1,"description":"Substitution into the integration by parts formula $uv - \\int v \\frac{du}{dx} \\, dx$.","code":"M"},{"id":"p3","marks":1,"description":"Integrates the remaining exponential term $e^x$ correctly.","code":"A"},{"id":"p4","marks":1,"description":"Correct final result with $+c$ included.","code":"A","acceptable":["x e^x - e^x + c","(x-1)e^x + c"]}],"finalAnswers":["x e^x - e^x + c"],"generatedAt":"2026-05-06T14:04:23.821Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"standard_formula","label":"Standard Formula","steps":[{"type":"text","order":0,"title":"Identify the formula","content":"To integrate $\\int x e^x \\, dx$, we use the standard **Integration by Parts** formula from your formula booklet:\n\n$$\\int u \\frac{dv}{dx} \\, dx = uv - \\int v \\frac{du}{dx} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Let } \\htmlData{anchor=u-choice}{\\color{@sky}{u}} = \\htmlData{anchor=u-val}{\\color{@sky}{x}} \\quad \\text{and} \\quad \\htmlData{anchor=dv-choice}{\\color{@amber}{\\frac{dv}{dx}}} = \\htmlData{anchor=dv-val}{\\color{@amber}{e^x}}$"},{"latex":"$$\\text{Then } \\htmlData{anchor=du-choice}{\\color{@purple}{\\frac{du}{dx}}} = \\htmlData{anchor=du-val}{\\color{@purple}{1}} \\quad \\text{and} \\quad \\htmlData{anchor=v-choice}{\\color{@teal}{v}} = \\htmlData{anchor=v-val}{\\color{@teal}{e^x}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-val"},{"color":"amber","style":"text-color","anchor":"dv-val"},{"color":"purple","style":"text-color","anchor":"du-val"},{"color":"teal","style":"text-color","anchor":"v-val"}],"connections":[{"to":"du-choice","from":"u-choice","color":"purple","label":"differentiate","style":"solid"},{"to":"v-choice","from":"dv-choice","color":"teal","label":"integrate","style":"solid"}]},"order":1,"title":"Choose $u$ and $v'$","description":"We choose $u = x$ because its derivative simplifies to $1$, and we choose $\\frac{dv}{dx} = e^x$ because its integral is easy to compute (it remains $e^x$). This earns the first method mark **M1**."},{"type":"text","order":2,"title":"Substitute into formula","content":"Now, substitute our terms $u$, $v$, and $\\frac{du}{dx}$ into the integration by parts formula:\n\n$$\\int x e^x \\, dx = x e^x - \\int e^x \\cdot 1 \\, dx$$\n\nThis simplifies directly to:\n\n$$x e^x - \\int e^x \\, dx$$\n\n*(This substitution earns the second method mark **M1**)*"},{"type":"text","order":3,"title":"Complete the integration","content":"Finally, we evaluate the remaining integral and add the constant of integration, $c$:\n\n$$\\int e^x \\, dx = e^x \\quad \\text{(earns accuracy mark } \\textbf{A1} \\text{)}$$\n\nPutting it all together gives our final answer:\n\n$$x e^x - e^x + c \\quad \\text{(earns accuracy mark } \\textbf{A1} \\text{)}$$\n\n*Tip: You can also write this in factored form as $e^x(x - 1) + c$.*"}]},{"id":"tabular_method","label":"Tabular Method","steps":[{"type":"text","order":0,"title":"Set up the table","content":"To solve $\\int x e^x \\, dx$ using the **Tabular Method** (also known as the DI method), we set up a table with three columns:\n\n1. **Sign**: Alternates starting with $+$.\n2. **D (Derivatives)**: The part of the integrand we will repeatedly differentiate. We choose $x$ because it differentiates to $0$ in a few steps.\n3. **I (Integrals)**: The part of the integrand we will repeatedly integrate. We choose $e^x$.\n\nLet's write down the initial row with the chosen functions."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\begin{array}{ccc} \\textbf{Sign} & \\textbf{D} & \\textbf{I} \\\\ \\hline \\htmlData{anchor=s1}{+} & \\htmlData{anchor=d1}{x} & \\htmlData{anchor=i1}{e^x} \\\\ \\htmlData{anchor=s2}{-} & \\htmlData{anchor=d2}{1} & \\htmlData{anchor=i2}{e^x} \\\\ \\htmlData{anchor=s3}{+} & \\htmlData{anchor=d3}{0} & \\htmlData{anchor=i3}{e^x} \\end{array}$$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"d1"},{"color":"amber","style":"text-color","anchor":"d2"},{"color":"purple","style":"text-color","anchor":"i1"},{"color":"purple","style":"text-color","anchor":"i2"}],"connections":[{"to":"d2","from":"d1","color":"amber","label":"differentiate","style":"solid"},{"to":"i2","from":"i1","color":"purple","label":"integrate","style":"solid"}]},"order":1,"title":"Fill in the rows","description":"We repeatedly differentiate the $D$ column and integrate the $I$ column until the derivative column reaches $0$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\begin{array}{ccc} \\textbf{Sign} & \\textbf{D} & \\textbf{I} \\\\ \\hline \\htmlData{anchor=sign1}{+} & \\htmlData{anchor=func1}{x} & e^x \\\\ \\htmlData{anchor=sign2}{-} & \\htmlData{anchor=func2}{1} & \\htmlData{anchor=int1}{e^x} \\\\ + & 0 & \\htmlData{anchor=int2}{e^x} \\end{array}$$"},{"latex":"$$$\\text{First product: } \\htmlData{anchor=p1}{\\color{@sky}{+(x)(e^x)}}$$"},{"latex":"$$$\\text{Second product: } \\htmlData{anchor=p2}{\\color{@pink}{-(1)(e^x)}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sign1"},{"color":"sky","style":"text-color","anchor":"func1"},{"color":"sky","style":"text-color","anchor":"int1"},{"color":"pink","style":"text-color","anchor":"sign2"},{"color":"pink","style":"text-color","anchor":"func2"},{"color":"pink","style":"text-color","anchor":"int2"}],"connections":[{"to":"int1","from":"func1","color":"sky","label":"multiply diagonal","style":"solid"},{"to":"int2","from":"func2","color":"pink","label":"multiply diagonal","style":"solid"}]},"order":2,"title":"Multiply along the diagonals","description":"To find the integral, we multiply each term in the $D$ column by the term below and to the right in the $I$ column, keeping the sign from that row."},{"type":"text","order":3,"title":"Sum the results","content":"Now we sum the diagonal products and add the constant of integration, $c$:\n\n$$ \\int x e^x \\, dx = x e^x - e^x + c $$\n\nWe can factor out $e^x$ to write it in its simplest form:\n\n$$ = e^x(x - 1) + c $$"}]}],"generatedAt":"2026-06-09T08:41:09.723Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"9974450d-b64e-4b3e-8e35-11698c71cdff:standard_formula","label":"Standard Formula","steps":[{"type":"text","order":0,"title":"Identify the formula","content":"To integrate $\\int x e^x \\, dx$, we use the standard **Integration by Parts** formula from your formula booklet:\n\n$$\\int u \\frac{dv}{dx} \\, dx = uv - \\int v \\frac{du}{dx} \\, dx$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Let } \\htmlData{anchor=u-choice}{\\color{@sky}{u}} = \\htmlData{anchor=u-val}{\\color{@sky}{x}} \\quad \\text{and} \\quad \\htmlData{anchor=dv-choice}{\\color{@amber}{\\frac{dv}{dx}}} = \\htmlData{anchor=dv-val}{\\color{@amber}{e^x}}$"},{"latex":"$$\\text{Then } \\htmlData{anchor=du-choice}{\\color{@purple}{\\frac{du}{dx}}} = \\htmlData{anchor=du-val}{\\color{@purple}{1}} \\quad \\text{and} \\quad \\htmlData{anchor=v-choice}{\\color{@teal}{v}} = \\htmlData{anchor=v-val}{\\color{@teal}{e^x}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-val"},{"color":"amber","style":"text-color","anchor":"dv-val"},{"color":"purple","style":"text-color","anchor":"du-val"},{"color":"teal","style":"text-color","anchor":"v-val"}],"connections":[{"to":"du-choice","from":"u-choice","color":"purple","label":"differentiate","style":"solid"},{"to":"v-choice","from":"dv-choice","color":"teal","label":"integrate","style":"solid"}]},"order":1,"title":"Choose $u$ and $v'$","description":"We choose $u = x$ because its derivative simplifies to $1$, and we choose $\\frac{dv}{dx} = e^x$ because its integral is easy to compute (it remains $e^x$). This earns the first method mark **M1**."},{"type":"text","order":2,"title":"Substitute into formula","content":"Now, substitute our terms $u$, $v$, and $\\frac{du}{dx}$ into the integration by parts formula:\n\n$$\\int x e^x \\, dx = x e^x - \\int e^x \\cdot 1 \\, dx$$\n\nThis simplifies directly to:\n\n$$x e^x - \\int e^x \\, dx$$\n\n*(This substitution earns the second method mark **M1**)*"},{"type":"text","order":3,"title":"Complete the integration","content":"Finally, we evaluate the remaining integral and add the constant of integration, $c$:\n\n$$\\int e^x \\, dx = e^x \\quad \\text{(earns accuracy mark } \\textbf{A1} \\text{)}$$\n\nPutting it all together gives our final answer:\n\n$$x e^x - e^x + c \\quad \\text{(earns accuracy mark } \\textbf{A1} \\text{)}$$\n\n*Tip: You can also write this in factored form as $e^x(x - 1) + c$.*"}]},{"id":"9974450d-b64e-4b3e-8e35-11698c71cdff:tabular_method","label":"Tabular Method","steps":[{"type":"text","order":0,"title":"Set up the table","content":"To solve $\\int x e^x \\, dx$ using the **Tabular Method** (also known as the DI method), we set up a table with three columns:\n\n1. **Sign**: Alternates starting with $+$.\n2. **D (Derivatives)**: The part of the integrand we will repeatedly differentiate. We choose $x$ because it differentiates to $0$ in a few steps.\n3. **I (Integrals)**: The part of the integrand we will repeatedly integrate. We choose $e^x$.\n\nLet's write down the initial row with the chosen functions."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\begin{array}{ccc} \\textbf{Sign} & \\textbf{D} & \\textbf{I} \\\\ \\hline \\htmlData{anchor=s1}{+} & \\htmlData{anchor=d1}{x} & \\htmlData{anchor=i1}{e^x} \\\\ \\htmlData{anchor=s2}{-} & \\htmlData{anchor=d2}{1} & \\htmlData{anchor=i2}{e^x} \\\\ \\htmlData{anchor=s3}{+} & \\htmlData{anchor=d3}{0} & \\htmlData{anchor=i3}{e^x} \\end{array}$$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"d1"},{"color":"amber","style":"text-color","anchor":"d2"},{"color":"purple","style":"text-color","anchor":"i1"},{"color":"purple","style":"text-color","anchor":"i2"}],"connections":[{"to":"d2","from":"d1","color":"amber","label":"differentiate","style":"solid"},{"to":"i2","from":"i1","color":"purple","label":"integrate","style":"solid"}]},"order":1,"title":"Fill in the rows","description":"We repeatedly differentiate the $D$ column and integrate the $I$ column until the derivative column reaches $0$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\begin{array}{ccc} \\textbf{Sign} & \\textbf{D} & \\textbf{I} \\\\ \\hline \\htmlData{anchor=sign1}{+} & \\htmlData{anchor=func1}{x} & e^x \\\\ \\htmlData{anchor=sign2}{-} & \\htmlData{anchor=func2}{1} & \\htmlData{anchor=int1}{e^x} \\\\ + & 0 & \\htmlData{anchor=int2}{e^x} \\end{array}$$"},{"latex":"$$$\\text{First product: } \\htmlData{anchor=p1}{\\color{@sky}{+(x)(e^x)}}$$"},{"latex":"$$$\\text{Second product: } \\htmlData{anchor=p2}{\\color{@pink}{-(1)(e^x)}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sign1"},{"color":"sky","style":"text-color","anchor":"func1"},{"color":"sky","style":"text-color","anchor":"int1"},{"color":"pink","style":"text-color","anchor":"sign2"},{"color":"pink","style":"text-color","anchor":"func2"},{"color":"pink","style":"text-color","anchor":"int2"}],"connections":[{"to":"int1","from":"func1","color":"sky","label":"multiply diagonal","style":"solid"},{"to":"int2","from":"func2","color":"pink","label":"multiply diagonal","style":"solid"}]},"order":2,"title":"Multiply along the diagonals","description":"To find the integral, we multiply each term in the $D$ column by the term below and to the right in the $I$ column, keeping the sign from that row."},{"type":"text","order":3,"title":"Sum the results","content":"Now we sum the diagonal products and add the constant of integration, $c$:\n\n$$ \\int x e^x \\, dx = x e^x - e^x + c $$\n\nWe can factor out $e^x$ to write it in its simplest form:\n\n$$ = e^x(x - 1) + c $$"}]}],"generatedAt":"2026-06-09T08:41:18.672Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"fe09803b-e565-4d2b-98d1-74682cd33551","revisionId":"2796eabe-f913-4736-9a2e-76d426d649e5","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"27222f06-2593-4816-975f-d99a8988502e","label":"a","content":"Use integration by parts to find $\\int 5t e^t \\, dt$","markscheme":"- Sets $u = 5t$ and $\\frac{dv}{dt} = e^t$, so $\\frac{du}{dt} = 5$ and $v = e^t$ M1\n- Applies formula: $5t e^t - \\int 5 e^t \\, dt$ M1\n- Integrates $5 e^t$ to get $5 e^t$ A1\n- Final answer: $5t e^t - 5 e^t + c$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Correct identification of $u$, $v$, $\\frac{du}{dt}$ and $\\frac{dv}{dt}$","code":"M"},{"id":"p2","marks":1,"description":"Correct substitution into the integration by parts formula using variable $t$","code":"M"},{"id":"p3","marks":1,"description":"Correct integration of the remaining exponential term","code":"A"},{"id":"p4","marks":1,"description":"Full correct expression including the constant of integration $+ c$","code":"A"}],"finalAnswers":["$$5t e^t - 5 e^t + c$"],"generatedAt":"2026-05-06T14:04:13.711Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_by_parts","label":"Direct Integration by Parts","steps":[{"type":"text","order":0,"title":"Recall the integration formula","content":"To integrate a product of two functions, we use the integration by parts formula from our data booklet:\n\n$$\\int u \\frac{dv}{dt} \\, dt = uv - \\int v \\frac{du}{dt} \\, dt$$\n\nOur goal is to choose $u$ and $\\frac{dv}{dt}$ such that the remaining integral $\\int v \\frac{du}{dt} \\, dt$ is easier to solve than the original one."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = \\htmlData{anchor=u-term}{5t} \\quad \\implies \\quad \\frac{du}{dt} = \\htmlData{anchor=du-term}{5}$"},{"latex":"$$\\frac{dv}{dt} = \\htmlData{anchor=dv-term}{e^t} \\quad \\implies \\quad v = \\htmlData{anchor=v-term}{e^t}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-term"},{"color":"sky","style":"text-color","anchor":"du-term"},{"color":"amber","style":"text-color","anchor":"dv-term"},{"color":"amber","style":"text-color","anchor":"v-term"}],"connections":[{"to":"du-term","from":"u-term","color":"sky","label":"differentiate","style":"solid"},{"to":"v-term","from":"dv-term","color":"amber","label":"integrate","style":"solid"}]},"order":1,"title":"Define components and differentiate","description":"We choose $u = 5t$ because differentiating it simplifies it to a constant. We choose $\\frac{dv}{dt} = e^t$ because it is straightforward to integrate. Under the markscheme, correctly identifying these parts earns you the first method mark (M1)."},{"type":"text","order":2,"title":"Apply integration by parts","content":"Now we substitute our values of $u$, $v$, $\\frac{du}{dt}$, and $\\frac{dv}{dt}$ into the integration by parts formula:\n\n$$\\int 5t e^t \\, dt = (5t)(e^t) - \\int (e^t)(5) \\, dt$$\n\nSimplifying this gives:\n\n$$5t e^t - \\int 5 e^t \\, dt$$\n\nWriting this down correctly earns you the second method mark (M1)."},{"type":"text","order":3,"title":"Perform the final integration","content":"Now, we integrate the remaining term. Since the integral of $e^t$ with respect to $t$ is simply $e^t$:\n\n$$\\int 5 e^t \\, dt = 5 e^t$$\n\nThis integration step earns you an accuracy mark (A1)."},{"type":"text","order":4,"title":"State the final expression","content":"Combining everything and adding the constant of integration $c$, we get:\n\n$$5t e^t - 5 e^t + c$$\n\nAlternatively, you can factor out $5e^t$ to write it as:\n\n$$5e^t(t - 1) + c$$\n\nBoth forms are correct and earn the final accuracy mark (A1)."}]},{"id":"factoring_constant","label":"Factoring out the Constant First","steps":[{"type":"text","order":0,"title":"Factor out the constant","content":"To simplify the integration, we can begin by factoring the constant coefficient $5$ out of the integral:\n\n$$\\int 5t e^t \\, dt = 5 \\int t e^t \\, dt$$","highlights":[{"text":"integration by parts to find \\int 5t e^t \\, dt","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"For the inner integral $\\int t e^t \\, dt$, we use the integration by parts formula:"},{"latex":"$$$\\int u \\frac{dv}{dt} \\, dt = uv - \\int v \\frac{du}{dt} \\, dt$$"},{"text":"Let us choose our parts $u$ and $\\frac{dv}{dt}$:"},{"latex":"$$$\\htmlData{anchor=u-choice}{\\color{@sky}{u = t}} \\implies \\htmlData{anchor=du-result}{\\color{@sky}{\\frac{du}{dt} = 1}}$$"},{"latex":"$$$\\htmlData{anchor=dv-choice}{\\color{@amber}{\\frac{dv}{dt} = e^t}} \\implies \\htmlData{anchor=v-result}{\\color{@amber}{v = e^t}}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"u-choice"},{"color":"amber","style":"box","anchor":"dv-choice"}],"connections":[{"to":"du-result","from":"u-choice","color":"sky","label":"differentiate","style":"solid"},{"to":"v-result","from":"dv-choice","color":"amber","label":"integrate","style":"solid"}]},"order":1,"title":"Set up Integration by Parts","description":"Using the LIATE rule, we choose the algebraic term $t$ as $u$ and the exponential term $e^t$ as $dv/dt$."},{"type":"text","order":2,"title":"Apply the integration formula","content":"Substitute $u$, $v$, $\\frac{du}{dt}$, and $\\frac{dv}{dt}$ into the integration by parts formula:\n\n$$\\int t e^t \\, dt = t e^t - \\int (e^t \\cdot 1) \\, dt$$\n\nEvaluating the remaining integral:\n\n$$\\int e^t \\, dt = e^t$$\n\nSo we get:\n\n$$\\int t e^t \\, dt = t e^t - e^t$$"},{"type":"text","order":3,"title":"Multiply by the constant","content":"Now, we multiply the entire integrated expression by the constant factor $5$ we separated in Step 1, and add the constant of integration $c$:\n\n$$\\begin{aligned} \\int 5t e^t \\, dt &= 5 \\left(t e^t - e^t\\right) + c \\\\ &= 5t e^t - 5e^t + c \\end{aligned}$$\n\n**Markscheme Alignment:**\n- Setting up the correct parts $u$ and $v$ earns **M1**.\n- Applying the parts formula correctly to reach the subtraction step earns **M1**.\n- Evaluating the second integral earns **A1**.\n- Stating the final simplified expression with $+ c$ earns **A1**."}]},{"id":"tabular_method","label":"Tabular Method","steps":[{"type":"text","order":0,"title":"Set up the DI Table","content":"The tabular method (or DI method) is a very efficient way to perform integration by parts, especially when one of the factors is a polynomial that simplifies to $0$ after repeated differentiation.\n\nWe set up a table with three columns:\n1. **Sign**: Alternating signs starting with $+$\n2. **D** (Differentiation): Choose the part that terminates upon differentiation, which is $5t$\n3. **I** (Integration): Choose the part that is easy to integrate, which is $e^t$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\begin{array}{ccc}\n\\text{\\bf Sign} & \\text{\\bf D} & \\text{\\bf I} \\\\\n\\hline\n\\htmlData{anchor=s1}{\\color{@sky}{+}} & \\htmlData{anchor=d1}{\\color{@amber}{5t}} & \\htmlData{anchor=i1}{\\color{@purple}{e^t}} \\\\\n\\htmlData{anchor=s2}{\\color{@sky}{-}} & \\htmlData{anchor=d2}{\\color{@amber}{5}} & \\htmlData{anchor=i2}{\\color{@purple}{e^t}} \\\\\n+ & 0 & e^t\n\\end{array}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"s1"},{"color":"amber","style":"text-color","anchor":"d1"},{"color":"purple","style":"text-color","anchor":"i1"}],"connections":[{"to":"i2","from":"d1","color":"teal","label":"multiply diagonally","style":"solid"}]},"order":1,"title":"Fill the DI table","description":"We differentiate $5t$ until we reach $0$, and integrate $e^t$ the same number of times. We also alternate the signs starting with $+1$."},{"type":"text","order":2,"title":"Multiply and combine terms","content":"We find the result by multiplying diagonally (each entry in the **D** column with the shifted next entry in the **I** column, multiplied by the corresponding sign in the **Sign** column):\n\n- First term: $$(+) \\cdot (5t) \\cdot (e^t) = 5t e^t$$\n- Second term: $$(-) \\cdot (5) \\cdot (e^t) = -5e^t$$"},{"type":"text","order":3,"title":"Write the final answer","content":"Summing the terms and adding the constant of integration $c$, we get:\n\n$$\\int 5t e^t \\, dt = 5t e^t - 5 e^t + c$$\n\nThis can also be written in factored form as:\n\n$$5e^t(t - 1) + c$$"}]}],"generatedAt":"2026-06-09T11:49:43.878Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"27222f06-2593-4816-975f-d99a8988502e:direct_by_parts","label":"Direct Integration by Parts","steps":[{"type":"text","order":0,"title":"Recall the integration formula","content":"To integrate a product of two functions, we use the integration by parts formula from our data booklet:\n\n$$\\int u \\frac{dv}{dt} \\, dt = uv - \\int v \\frac{du}{dt} \\, dt$$\n\nOur goal is to choose $u$ and $\\frac{dv}{dt}$ such that the remaining integral $\\int v \\frac{du}{dt} \\, dt$ is easier to solve than the original one."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = \\htmlData{anchor=u-term}{5t} \\quad \\implies \\quad \\frac{du}{dt} = \\htmlData{anchor=du-term}{5}$"},{"latex":"$$\\frac{dv}{dt} = \\htmlData{anchor=dv-term}{e^t} \\quad \\implies \\quad v = \\htmlData{anchor=v-term}{e^t}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-term"},{"color":"sky","style":"text-color","anchor":"du-term"},{"color":"amber","style":"text-color","anchor":"dv-term"},{"color":"amber","style":"text-color","anchor":"v-term"}],"connections":[{"to":"du-term","from":"u-term","color":"sky","label":"differentiate","style":"solid"},{"to":"v-term","from":"dv-term","color":"amber","label":"integrate","style":"solid"}]},"order":1,"title":"Define components and differentiate","description":"We choose $u = 5t$ because differentiating it simplifies it to a constant. We choose $\\frac{dv}{dt} = e^t$ because it is straightforward to integrate. Under the markscheme, correctly identifying these parts earns you the first method mark (M1)."},{"type":"text","order":2,"title":"Apply integration by parts","content":"Now we substitute our values of $u$, $v$, $\\frac{du}{dt}$, and $\\frac{dv}{dt}$ into the integration by parts formula:\n\n$$\\int 5t e^t \\, dt = (5t)(e^t) - \\int (e^t)(5) \\, dt$$\n\nSimplifying this gives:\n\n$$5t e^t - \\int 5 e^t \\, dt$$\n\nWriting this down correctly earns you the second method mark (M1)."},{"type":"text","order":3,"title":"Perform the final integration","content":"Now, we integrate the remaining term. Since the integral of $e^t$ with respect to $t$ is simply $e^t$:\n\n$$\\int 5 e^t \\, dt = 5 e^t$$\n\nThis integration step earns you an accuracy mark (A1)."},{"type":"text","order":4,"title":"State the final expression","content":"Combining everything and adding the constant of integration $c$, we get:\n\n$$5t e^t - 5 e^t + c$$\n\nAlternatively, you can factor out $5e^t$ to write it as:\n\n$$5e^t(t - 1) + c$$\n\nBoth forms are correct and earn the final accuracy mark (A1)."}]},{"id":"27222f06-2593-4816-975f-d99a8988502e:factoring_constant","label":"Factoring out the Constant First","steps":[{"type":"text","order":0,"title":"Factor out the constant","content":"To simplify the integration, we can begin by factoring the constant coefficient $5$ out of the integral:\n\n$$\\int 5t e^t \\, dt = 5 \\int t e^t \\, dt$$","highlights":[{"text":"integration by parts to find \\int 5t e^t \\, dt","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"For the inner integral $\\int t e^t \\, dt$, we use the integration by parts formula:"},{"latex":"$$$\\int u \\frac{dv}{dt} \\, dt = uv - \\int v \\frac{du}{dt} \\, dt$$"},{"text":"Let us choose our parts $u$ and $\\frac{dv}{dt}$:"},{"latex":"$$$\\htmlData{anchor=u-choice}{\\color{@sky}{u = t}} \\implies \\htmlData{anchor=du-result}{\\color{@sky}{\\frac{du}{dt} = 1}}$$"},{"latex":"$$$\\htmlData{anchor=dv-choice}{\\color{@amber}{\\frac{dv}{dt} = e^t}} \\implies \\htmlData{anchor=v-result}{\\color{@amber}{v = e^t}}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"u-choice"},{"color":"amber","style":"box","anchor":"dv-choice"}],"connections":[{"to":"du-result","from":"u-choice","color":"sky","label":"differentiate","style":"solid"},{"to":"v-result","from":"dv-choice","color":"amber","label":"integrate","style":"solid"}]},"order":1,"title":"Set up Integration by Parts","description":"Using the LIATE rule, we choose the algebraic term $t$ as $u$ and the exponential term $e^t$ as $dv/dt$."},{"type":"text","order":2,"title":"Apply the integration formula","content":"Substitute $u$, $v$, $\\frac{du}{dt}$, and $\\frac{dv}{dt}$ into the integration by parts formula:\n\n$$\\int t e^t \\, dt = t e^t - \\int (e^t \\cdot 1) \\, dt$$\n\nEvaluating the remaining integral:\n\n$$\\int e^t \\, dt = e^t$$\n\nSo we get:\n\n$$\\int t e^t \\, dt = t e^t - e^t$$"},{"type":"text","order":3,"title":"Multiply by the constant","content":"Now, we multiply the entire integrated expression by the constant factor $5$ we separated in Step 1, and add the constant of integration $c$:\n\n$$\\begin{aligned} \\int 5t e^t \\, dt &= 5 \\left(t e^t - e^t\\right) + c \\\\ &= 5t e^t - 5e^t + c \\end{aligned}$$\n\n**Markscheme Alignment:**\n- Setting up the correct parts $u$ and $v$ earns **M1**.\n- Applying the parts formula correctly to reach the subtraction step earns **M1**.\n- Evaluating the second integral earns **A1**.\n- Stating the final simplified expression with $+ c$ earns **A1**."}]},{"id":"27222f06-2593-4816-975f-d99a8988502e:tabular_method","label":"Tabular Method","steps":[{"type":"text","order":0,"title":"Set up the DI Table","content":"The tabular method (or DI method) is a very efficient way to perform integration by parts, especially when one of the factors is a polynomial that simplifies to $0$ after repeated differentiation.\n\nWe set up a table with three columns:\n1. **Sign**: Alternating signs starting with $+$\n2. **D** (Differentiation): Choose the part that terminates upon differentiation, which is $5t$\n3. **I** (Integration): Choose the part that is easy to integrate, which is $e^t$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\begin{array}{ccc}\n\\text{\\bf Sign} & \\text{\\bf D} & \\text{\\bf I} \\\\\n\\hline\n\\htmlData{anchor=s1}{\\color{@sky}{+}} & \\htmlData{anchor=d1}{\\color{@amber}{5t}} & \\htmlData{anchor=i1}{\\color{@purple}{e^t}} \\\\\n\\htmlData{anchor=s2}{\\color{@sky}{-}} & \\htmlData{anchor=d2}{\\color{@amber}{5}} & \\htmlData{anchor=i2}{\\color{@purple}{e^t}} \\\\\n+ & 0 & e^t\n\\end{array}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"s1"},{"color":"amber","style":"text-color","anchor":"d1"},{"color":"purple","style":"text-color","anchor":"i1"}],"connections":[{"to":"i2","from":"d1","color":"teal","label":"multiply diagonally","style":"solid"}]},"order":1,"title":"Fill the DI table","description":"We differentiate $5t$ until we reach $0$, and integrate $e^t$ the same number of times. We also alternate the signs starting with $+1$."},{"type":"text","order":2,"title":"Multiply and combine terms","content":"We find the result by multiplying diagonally (each entry in the **D** column with the shifted next entry in the **I** column, multiplied by the corresponding sign in the **Sign** column):\n\n- First term: $$(+) \\cdot (5t) \\cdot (e^t) = 5t e^t$$\n- Second term: $$(-) \\cdot (5) \\cdot (e^t) = -5e^t$$"},{"type":"text","order":3,"title":"Write the final answer","content":"Summing the terms and adding the constant of integration $c$, we get:\n\n$$\\int 5t e^t \\, dt = 5t e^t - 5 e^t + c$$\n\nThis can also be written in factored form as:\n\n$$5e^t(t - 1) + 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