tegration by substitution (A-level only) Flashcards","topicId":"7d4c8fb4-456a-4cfc-a419-05fa7441e2a1","topicSlug":"integration-by-substitution","cards":[{"id":"05b03f5c-f985-4e99-832c-008db27100bd","cardType":"basic","front":"What must be added after evaluating an indefinite integral?","back":"The constant of integration, $+C$.","points":1,"mcqOptions":null},{"id":"fa8c1e00-096c-4516-adb5-fa735f32bf3d","cardType":"basic","front":"Why is $+C$ not included after evaluating a definite integral?","back":"A definite integral gives a numerical value, not a family of antiderivatives.","points":2,"mcqOptions":null},{"id":"47eea841-b420-4ab5-9ac2-dd4f73234db8","cardType":"cloze","front":"If $u=g(x)$, then $\\text{[...]}$.","back":"If $u=g(x)$, then $\\boxed{du=g'(x)\\,dx}$.","points":null,"mcqOptions":null},{"id":"ece40f6e-336a-4cb9-be8b-c714648747c6","cardType":"basic","front":"What must be true of the transformed integral before integrating with respect to $u$?","back":"It must be written entirely in terms of $u$.","points":1,"mcqOptions":null},{"id":"0fd08738-425f-459c-a57a-f3c31dc45139","cardType":"basic","front":"Why is substitution useful when an integrand contains an inner function and its derivative?","back":"It reverses the chain rule.","points":2,"mcqOptions":null},{"id":"30a6509d-5581-4b42-8f92-44cbf2797852","cardType":"basic","front":"How should a constant mismatch between $du$ and the integrand be handled?","back":"Move the required constant factor outside the integral.","points":1,"mcqOptions":null},{"id":"ad3fd537-de3e-4582-8bc1-936669af2a51","cardType":"basic","front":"Evaluate $\\displaystyle \\int 6x(3x^2+1)^4\\,dx$.","back":"$\\displaystyle \\frac{1}{5}(3x^2+1)^5+C$","points":2,"mcqOptions":null},{"id":"f99c4c2e-7281-439a-9e8b-fb21fb2e03cc","cardType":"basic","front":"Evaluate $\\displaystyle \\int (4x-3)^6\\,dx$.","back":"$\\displaystyle \\frac{(4x-3)^7}{28}+C$","points":2,"mcqOptions":null},{"id":"e2aae55d-9849-4f35-9ae4-18f4ee3e0fe4","cardType":"basic","front":"For $u=x^2+4$, what replaces $x\\,dx$?","back":"$\\displaystyle x\\,dx=\\frac{1}{2}\\,du$","points":1,"mcqOptions":null},{"id":"c094a05b-05e5-47af-9e3a-2bf679dcf1ef","cardType":"basic","front":"Evaluate $\\displaystyle \\int x\\sqrt{x^2+4}\\,dx$.","back":"$\\displaystyle \\frac{1}{3}(x^2+4)^{3/2}+C$","points":2,"mcqOptions":null},{"id":"d97ff5d9-8ad1-4fbc-ab18-865f42134a5c","cardType":"cloze","front":"The reciprocal integral is $\\text{[...]}$.","back":"The reciprocal integral is $\\boxed{\\displaystyle \\int \\frac{1}{u}\\,du=\\ln|u|+C}$.","points":null,"mcqOptions":null},{"id":"8c205407-5233-46d1-b9fd-ae90e86c68d6","cardType":"basic","front":"Why are modulus signs needed in $\\ln|u|$?","back":"The reciprocal is defined for negative and positive $u$, excluding zero.","points":2,"mcqOptions":null},{"id":"46f064e4-2da1-446c-9b09-ecbd33456f7c","cardType":"basic","front":"Evaluate $\\displaystyle \\int \\frac{3x^2}{x^3-7}\\,dx$.","back":"$\\displaystyle \\ln|x^3-7|+C$","points":2,"mcqOptions":null},{"id":"8f610326-c25b-463c-a0e4-be7ada5908a2","cardType":"cloze","front":"Under $u=g(x)$, the limit $x=a$ becomes $\\text{[...]}$, while $x=b$ becomes $u=g(b)$.","back":"Under $u=g(x)$, the limit $x=a$ becomes $\\boxed{u=g(a)}$, while $x=b$ becomes $u=g(b)$.","points":null,"mcqOptions":null},{"id":"23539a7d-7393-485c-a59b-168e7fc43a41","cardType":"cloze","front":"Under $u=g(x)$, the limit $x=a$ becomes $u=g(a)$, while $x=b$ becomes $\\text{[...]}$.","back":"Under $u=g(x)$, the limit $x=a$ becomes $u=g(a)$, while $x=b$ becomes $\\boxed{u=g(b)}$.","points":null,"mcqOptions":null},{"id":"1f78fccf-54fb-4615-84d2-b0e9c5927738","cardType":"basic","front":"Under $u=2x+1$, what are the

1.8.8 Integration by substitution (A-level only) Flashcards

1.8.8 Integration by substitution (A-level only) Flashcards