shcards","topicId":"53edcf23-243a-46a3-8a7b-03d789a50f35","topicSlug":"integration-6","cards":[{"id":"6121121e-843d-483d-9905-995c4203473b","cardType":"basic","front":"Why must an indefinite integral include $+C$?","back":"Every constant differentiates to zero.","points":2,"mcqOptions":null},{"id":"4ab110c1-cb2e-468c-9b8c-b485d5e77258","cardType":"basic","front":"How can you check a proposed indefinite integral?","back":"Differentiate it and recover the integrand exactly.","points":1,"mcqOptions":null},{"id":"8f4b60f2-5b4e-458c-9f48-bb7dfab19b2a","cardType":"basic","front":"Why is $e^{kx}$ divided by $k$ when integrated?","back":"Differentiating $e^{kx}$ introduces the factor $k$.","points":2,"mcqOptions":null},{"id":"0f1590ba-f5da-4046-85e2-68696d78c4cf","cardType":"cloze","front":"For $k\\neq0$, $\\displaystyle \\int e^{kx}\\,dx=$ $\\text{[...]}$.","back":"For $k\\neq0$, $\\displaystyle \\int e^{kx}\\,dx=$ $\\boxed{\\displaystyle \\frac{1}{k}e^{kx}+C}$.","points":null,"mcqOptions":null},{"id":"c240cf72-a3a8-47e5-8a98-2680a6528d0e","cardType":"cloze","front":"For $x\\neq0$, $\\displaystyle \\int \\frac{1}{x}\\,dx=$ $\\text{[...]}$.","back":"For $x\\neq0$, $\\displaystyle \\int \\frac{1}{x}\\,dx=$ $\\boxed{\\displaystyle \\ln|x|+C}$.","points":null,"mcqOptions":null},{"id":"ad44d965-5a4c-44fa-9e4b-b1b1a7268865","cardType":"basic","front":"Why is the modulus needed in $\\ln|x|$?","back":"It makes the antiderivative valid where $x<0$ or $x>0$.","points":2,"mcqOptions":null},{"id":"c47b9628-3f99-4cfe-82a1-5a20be001812","cardType":"basic","front":"Why can an antiderivative of $\\frac{1}{x}$ not apply across $x=0$?","back":"The integrand is undefined at $x=0$.","points":1,"mcqOptions":null},{"id":"ef4ac41a-f4d0-4199-baf8-f8461ef19e6d","cardType":"cloze","front":"For $k\\neq0$, $\\displaystyle \\int \\cos(kx)\\,dx=$ $\\text{[...]}$.","back":"For $k\\neq0$, $\\displaystyle \\int \\cos(kx)\\,dx=$ $\\boxed{\\displaystyle \\frac{1}{k}\\sin(kx)+C}$.","points":null,"mcqOptions":null},{"id":"709582c6-dec6-4ad1-bf37-11ea99956fbd","cardType":"cloze","front":"For $k\\neq0$, $\\displaystyle \\int \\sin(kx)\\,dx=$ $\\text{[...]}$.","back":"For $k\\neq0$, $\\displaystyle \\int \\sin(kx)\\,dx=$ $\\boxed{\\displaystyle -\\frac{1}{k}\\cos(kx)+C}$.","points":null,"mcqOptions":null},{"id":"a31dd891-770b-4538-9489-dd1fbb43be3b","cardType":"basic","front":"What sign change occurs when integrating $\\sin(kx)$?","back":"It becomes negative cosine.","points":1,"mcqOptions":null},{"id":"f3fc79bc-44cb-464a-9e51-5484c80118fa","cardType":"basic","front":"Find $\\displaystyle \\int 5e^{3x}\\,dx$.","back":"$\\displaystyle \\frac{5}{3}e^{3x}+C$","points":1,"mcqOptions":null},{"id":"7e5f062c-b237-42b4-b03c-fe750eb23667","cardType":"basic","front":"Find $\\displaystyle \\int 4\\cos(2x)\\,dx$.","back":"$\\displaystyle 2\\sin(2x)+C$","points":1,"mcqOptions":null},{"id":"94b35aed-6dc2-43ab-afd6-ab8691705c03","cardType":"basic","front":"Find $\\displaystyle \\int -3\\sin(5x)\\,dx$.","back":"$\\displaystyle \\frac{3}{5}\\cos(5x)+C$","points":1,"mcqOptions":null},{"id":"d25f9200-26e9-4c0d-9d79-e0c7ebda7176","cardType":"cloze","front":"The reverse-chain-rule pattern $\\displaystyle \\int f'(x)e^{f(x)}\\,dx$ gives $\\text{[...]}$.","back":"The reverse-chain-rule pattern $\\displaystyle \\int f'(x)e^{f(x)}\\,dx$ gives $\\boxed{\\displaystyle e^{f(x)}+C}$.","points":null,"mcqOptions":null},{"id":"c59b936e-1056-4bca-b0a8-cbe5f12a0e35","cardType":"basic","front":"Find $\\displaystyle \\int xe^{x^2}\\,dx$.","back":"$\\displaystyle \\frac{1}{2}e^{x^2}+C$","points":2,"mcqOptions":null},{"id":"bafebbd2-a3b6-4741-af56-82ecafc64f51","cardType":"basic","front":"What feature suggests an integral will involve $\\ln|f(x)|$?","back":"The numerator is a constant multiple of $f'(x)$.","points":1,"mcqOptions":null},{"id":"8fee36ec-355e-4791-aea8-b6bee20adb2f","cardType":"cloze","front":"Provide

3.6.1 Integration (A-level only) Flashcards

3.6.1 Integration (A-level only) Flashcards