Question

describe the method used to integrate sin³x. choose the correct answer below. a. rewrite sin³x as tan³x cos³x, then use the substitution u = cos x. b. rewrite sin³x as (1 - cos²x) sin x, then use a half - angle formula to rewrite the cos²x term. c. rewrite sin³x as (sin²x) sin x, then use a half - angle formula to rewrite the sin²x term. d. rewrite sin³x as (1 - cos²x) sin x, then use the substitution u = cos x.

Explanation:

Step1: Recall trigonometric identity

We know that $\sin^{2}x=1 - \cos^{2}x$, so $\sin^{3}x=\sin^{2}x\sin x=(1 - \cos^{2}x)\sin x$.

Step2: Consider substitution method

If we let $u = \cos x$, then $du=-\sin xdx$. The integral $\int\sin^{3}x dx=\int(1 - \cos^{2}x)\sin xdx$ can be rewritten in terms of $u$ as $-\int(1 - u^{2})du$ after substitution.

Answer:

D. Rewrite $\sin^{3}x$ as $(1 - \cos^{2}x)\sin x$, then use the substitution $u = \cos x$.