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Problem 5 - Entrance Test
Evaluate ∫ sin^3(x) cos^2(x) dx.
A. (1/5)cos^5(x) - (1/3)cos^3(x) + C
B. -(1/5)cos^5(x) + (1/3)cos^3(x) + C
C. (1/5)sin^5(x) - (1/3)sin^3(x) + C
D. -(1/5)sin^5(x) + (1/3)sin^3(x) + C
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Correct: A
∫ sin^3(x) cos^2(x) dx = ∫ sin^2(x) cos^2(x) sin(x) dx = ∫ (1 - cos^2(x)) cos^2(x) sin(x) dx. Let u = cos(x), du = -sin(x) dx. Then ∫ (1 - u^2) u^2 (-du) = ∫ (u^4 - u^2) du = (1/5)u^5 - (1/3)u^3 + C = (1/5)cos^5(x) - (1/3)cos^3(x) + C.