Question

Integrate. Choose the best approach and the answer. LaTeX: \int\sin^3x\:dx ∫ sin 3 ⁡ x d x a. use LaTeX: \sin^2x=\frac{1}{2}\left(1-\cos2x\right) sin 2 ⁡ x = 1 2 ( 1 − cos ⁡ 2 x ) , then use u-substitution b. use LaTeX: \sin^2x=1-\cos^2x sin 2 ⁡ x = 1 − cos 2 ⁡ x , then use u-substitution c. LaTeX: -\cos x+\frac{1}{3}\cos^3x\:+C − cos ⁡ x + 1 3 cos 3 ⁡ x + C d. LaTeX: \frac{1}{3}\cos^3x\:+C 1 3 cos 3 ⁡ x + C e. LaTeX: \frac{1}{3}\cos^3x-\frac{2}{3}\sin^3x\:+C

Answer 1

Reduce the power by applying the identity,

$\sin^2x+\cos^2x=1$

$\implies\displaystyle\int\sin^3x\,\mathrm dx=\int\sin x(1-\cos^2x)\,\mathrm dx$

Let $u=\cos x\implies\mathrm du=-\sin x\,\mathrm dx$:

$\implies\displaystyle\int\sin^3x\,\mathrm dx=-\int(1-u^2)\,\mathrm du$

$=\dfrac{u^3}3-u+C=\boxed{\dfrac{\cos^3x}3-\cos x+C}$