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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":42,"initialQuestionWindow":[{"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) + c$$"}]}],"generatedAt":"2026-06-09T11:49:43.920Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"5a1c80f0-652d-4d35-a828-03ec3eace025","revisionId":"dd1516a4-4fe2-4abe-a535-6451d1114321","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"ffe89a9d-b428-473f-b05c-29a8c781c1d7","label":"a","content":"Use the substitution $u = \\tan x$ to find $\\int \\tan^5 x \\sec^2 x dx$","markscheme":"Let $u = \\tan x \\implies \\frac{du}{dx} = \\sec^2 x$ M1\nSubstitute: $\\int u^5 \\sec^2 x \\cdot \\frac{1}{\\sec^2 x} du = \\int u^5 du$ M1\nIntegrate: $\\frac{1}{6}u^6 + C$ M1\nSubstitute back: $\\frac{1}{6}\\tan^6 x + C$ A2","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Differentiates substitution to find du/dx.","code":"M"},{"id":"p2","marks":1,"description":"Correctly substitutes into the integral to obtain a function of u only.","code":"M"},{"id":"p3","marks":1,"description":"Integrates the power of u correctly.","code":"M"},{"id":"p4","marks":2,"description":"Provides the final back-substituted expression with constant of integration.","code":"A","acceptable":["1/6 tan^6(x) + C","\\frac{\\tan^6 x}{6} + C"]}],"finalAnswers":["\\frac{1}{6}\\tan^6 x + C"],"generatedAt":"2026-05-06T14:04:28.557Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"formal_u_substitution","label":"Formal u-Substitution","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Let } \\htmlData{anchor=u-def}{\\color{@sky}{u = \\tan x}}$"},{"latex":"$$\\frac{du}{dx} = \\htmlData{anchor=du-deriv}{\\color{@amber}{\\sec^2 x}}$"},{"latex":"$$\\htmlData{anchor=dx-expr}{\\color{@purple}{dx = \\frac{du}{\\sec^2 x}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-def"},{"color":"amber","style":"text-color","anchor":"du-deriv"}],"connections":[{"to":"du-deriv","from":"u-def","color":"amber","label":"differentiate","style":"solid"}]},"order":0,"title":"Define the substitution","description":"We start by setting $u = \\tan x$. Differentiating both sides with respect to $x$ gives $\\frac{du}{dx} = \\sec^2 x$, which we rearrange to express $dx$ in terms of $du$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int \\htmlData{anchor=orig-tan}{\\color{@sky}{\\tan^5 x}} \\sec^2 x \\, \\htmlData{anchor=orig-dx}{\\color{@purple}{dx}}$"},{"latex":"$$= \\int \\htmlData{anchor=sub-u}{\\color{@sky}{u^5}} \\sec^2 x \\cdot \\htmlData{anchor=sub-dx}{\\color{@purple}{\\frac{du}{\\sec^2 x}}}$"},{"latex":"$$= \\int u^5 \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-u"},{"color":"purple","style":"text-color","anchor":"sub-dx"}],"connections":[{"to":"sub-u","from":"orig-tan","color":"sky","label":"u = tan x","style":"solid"},{"to":"sub-dx","from":"orig-dx","color":"purple","label":"substitute dx","style":"solid"}]},"order":1,"title":"Substitute into the integral","description":"Now, replace $\\tan x$ with $u$ and $dx$ with $\\frac{du}{\\sec^2 x}$. The $\\sec^2 x$ terms cancel out, leaving a simple integral in terms of $u$."},{"type":"text","order":2,"title":"Integrate with respect to u","content":"Using the power rule for integration, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1} + C$, we integrate $u^5$:\n\n$$\\int u^5 \\, du = \\frac{1}{6}u^6 + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{1}{6} \\htmlData{anchor=u-term}{\\color{@sky}{u}}^6 + C$"},{"latex":"$$= \\frac{1}{6} (\\htmlData{anchor=tan-term}{\\color{@sky}{\\tan x}})^6 + C$"},{"latex":"$$= \\frac{1}{6}\\tan^6 x + C$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-term"},{"color":"sky","style":"text-color","anchor":"tan-term"}],"connections":[{"to":"tan-term","from":"u-term","color":"sky","label":"substitute back","style":"solid"}]},"order":3,"title":"Substitute back to x","description":"Finally, replace $u$ with $\\tan x$ to write the final answer in terms of $x$. Don't forget to include the constant of integration, $C$."}]}],"generatedAt":"2026-06-09T05:13:36.924Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"ffe89a9d-b428-473f-b05c-29a8c781c1d7:formal_u_substitution","label":"Formal u-Substitution","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Let } \\htmlData{anchor=u-def}{\\color{@sky}{u = \\tan x}}$"},{"latex":"$$\\frac{du}{dx} = \\htmlData{anchor=du-deriv}{\\color{@amber}{\\sec^2 x}}$"},{"latex":"$$\\htmlData{anchor=dx-expr}{\\color{@purple}{dx = \\frac{du}{\\sec^2 x}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-def"},{"color":"amber","style":"text-color","anchor":"du-deriv"}],"connections":[{"to":"du-deriv","from":"u-def","color":"amber","label":"differentiate","style":"solid"}]},"order":0,"title":"Define the substitution","description":"We start by setting $u = \\tan x$. Differentiating both sides with respect to $x$ gives $\\frac{du}{dx} = \\sec^2 x$, which we rearrange to express $dx$ in terms of $du$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int \\htmlData{anchor=orig-tan}{\\color{@sky}{\\tan^5 x}} \\sec^2 x \\, \\htmlData{anchor=orig-dx}{\\color{@purple}{dx}}$"},{"latex":"$$= \\int \\htmlData{anchor=sub-u}{\\color{@sky}{u^5}} \\sec^2 x \\cdot \\htmlData{anchor=sub-dx}{\\color{@purple}{\\frac{du}{\\sec^2 x}}}$"},{"latex":"$$= \\int u^5 \\, du$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sub-u"},{"color":"purple","style":"text-color","anchor":"sub-dx"}],"connections":[{"to":"sub-u","from":"orig-tan","color":"sky","label":"u = tan x","style":"solid"},{"to":"sub-dx","from":"orig-dx","color":"purple","label":"substitute dx","style":"solid"}]},"order":1,"title":"Substitute into the integral","description":"Now, replace $\\tan x$ with $u$ and $dx$ with $\\frac{du}{\\sec^2 x}$. The $\\sec^2 x$ terms cancel out, leaving a simple integral in terms of $u$."},{"type":"text","order":2,"title":"Integrate with respect to u","content":"Using the power rule for integration, $\\int u^n \\, du = \\frac{u^{n+1}}{n+1} + C$, we integrate $u^5$:\n\n$$\\int u^5 \\, du = \\frac{1}{6}u^6 + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{1}{6} \\htmlData{anchor=u-term}{\\color{@sky}{u}}^6 + C$"},{"latex":"$$= \\frac{1}{6} (\\htmlData{anchor=tan-term}{\\color{@sky}{\\tan x}})^6 + C$"},{"latex":"$$= \\frac{1}{6}\\tan^6 x + C$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"u-term"},{"color":"sky","style":"text-color","anchor":"tan-term"}],"connections":[{"to":"tan-term","from":"u-term","color":"sky","label":"substitute back","style":"solid"}]},"order":3,"title":"Substitute back to x","description":"Finally, replace $u$ with $\\tan x$ to write the final answer in terms of $x$. Don't forget to include the constant of integration, 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Integration (A-level only)","topicId":"42968cfd-fa73-4774-88bc-80d554297685","topicSidebar":"$L3c","topicSlug":"integration-9","topicTitle":"3.6.4 Integration (A-level only)"}]}],"$L3d","$L3e"]}]

3.6.4 Integration (A-level only)

Question 5

3.6.4 Integration (A-level only) Questions

3.6.4 Integration (A-level only)

Question 5

3.6.4 Integration (A-level only) Questions