arts (A-level only) Flashcards","topicId":"fc25d0e0-d884-4857-8a61-bf5eb25cd695","topicSlug":"integration-by-parts","cards":[{"id":"809ed763-c960-4479-819d-e71197606475","cardType":"basic","front":"What should happen to $u$ when it is differentiated?","back":"It should become simpler.","points":1,"mcqOptions":null},{"id":"b14cf476-a9c1-4fad-8bc0-add6401d325f","cardType":"basic","front":"What property should $\\frac{dv}{dx}$ have?","back":"It should be convenient to integrate.","points":1,"mcqOptions":null},{"id":"f9573052-70da-4ff4-8fe8-d588278f9344","cardType":"cloze","front":"The integration by parts formula is $\\int u\\,dv =$ $\\text{[...]}$ $-$ $\\int v\\,du$.","back":"The integration by parts formula is $\\int u\\,dv =$ $\\boxed{uv}$ $-$ $\\int v\\,du$.","points":null,"mcqOptions":null},{"id":"9f61c811-bb27-44f9-a5db-7a85a48d52f3","cardType":"cloze","front":"The integration by parts formula is $\\int u\\,dv =$ $uv$ $-$ $\\text{[...]}$.","back":"The integration by parts formula is $\\int u\\,dv =$ $uv$ $-$ $\\boxed{\\int v\\,du}$.","points":null,"mcqOptions":null},{"id":"82ddbb99-a189-45f0-ba3e-1d321cda06b2","cardType":"basic","front":"For $\\int xe^x\\,dx$, why is $u=x$ a sensible choice?","back":"Differentiating $x$ gives the simpler function $1$.","points":2,"mcqOptions":null},{"id":"920522a8-8940-43ae-99a8-11d031124025","cardType":"basic","front":"How can $\\int \\ln x\\,dx$ be treated as an integral of a product?","back":"Rewrite the integrand as $\\ln x\\times 1$.","points":1,"mcqOptions":null},{"id":"577f3494-c12a-484b-b116-733fd89a395c","cardType":"basic","front":"Evaluate $\\int xe^x\\,dx$.","back":"$e^x(x-1)+C$","points":2,"mcqOptions":null},{"id":"a622af50-e9ad-4a16-9f27-36ea16993f9f","cardType":"basic","front":"Evaluate $\\int x\\cos x\\,dx$.","back":"$x\\sin x+\\cos x+C$","points":2,"mcqOptions":null},{"id":"88c2c318-afed-4cbd-93c8-ff8a729a8357","cardType":"basic","front":"Evaluate $\\int \\ln x\\,dx$ for $x>0$.","back":"$x\\ln x-x+C$","points":2,"mcqOptions":null},{"id":"aad65265-5efc-4739-b9a7-f180e406b98b","cardType":"cloze","front":"For definite integrals, $\\int_a^b u\\frac{dv}{dx}\\,dx =$ $\\text{[...]}$ $-$ $\\int_a^b v\\frac{du}{dx}\\,dx$.","back":"For definite integrals, $\\int_a^b u\\frac{dv}{dx}\\,dx =$ $\\boxed{[uv]_a^b}$ $-$ $\\int_a^b v\\frac{du}{dx}\\,dx$.","points":null,"mcqOptions":null},{"id":"be47446a-ee2f-4c4a-bf55-9565376f3186","cardType":"cloze","front":"For definite integrals, $\\int_a^b u\\frac{dv}{dx}\\,dx =$ $[uv]_a^b$ $-$ $\\text{[...]}$.","back":"For definite integrals, $\\int_a^b u\\frac{dv}{dx}\\,dx =$ $[uv]_a^b$ $-$ $\\boxed{\\int_a^b v\\frac{du}{dx}\\,dx}$.","points":null,"mcqOptions":null},{"id":"ff335421-d1cb-4ce9-a570-f7ecdbdb908e","cardType":"basic","front":"What does $[uv]_a^b$ mean?","back":"$uv$ at $b$ minus $uv$ at $a$.","points":1,"mcqOptions":null},{"id":"a6c5f259-1802-4bed-ba0f-fa2dfb8b04c4","cardType":"basic","front":"Why are two applications of integration by parts expected for $\\int x^2e^x\\,dx$?","back":"Each differentiation reduces the degree of $x^2$ by one.","points":2,"mcqOptions":null},{"id":"28649ca2-a236-48c4-b1a4-b3ed1cb76bd8","cardType":"basic","front":"What does the first application of integration by parts give for $\\int x^2e^x\\,dx$?","back":"$x^2e^x-\\int 2xe^x\\,dx$","points":2,"mcqOptions":null},{"id":"def8035e-0180-4261-bffe-c6c3b09792de","cardType":"basic","front":"Evaluate $\\int x^2e^x\\,dx$.","back":"$e^x(x^2-2x+2)+C$","points":2,"mcqOptions":null},{"id":"b981b618-fb44-45eb-9a80-7b2041c89f1c","cardType":"basic","front":"Why are brackets essential when substituting a repeated integratio

1.8.9 Integration by parts (A-level only) Flashcards

1.8.9 Integration by parts (A-level only) Flashcards