Question

evaluate the indefinite integral. \\(\int 2\sin^3(x)\cos(x)dx = \square + c\\)

Explanation:

Step1: Choose substitution variable

Let $u = \sin(x)$

Step2: Compute derivative of u

$\frac{du}{dx} = \cos(x) \implies du = \cos(x)dx$

Step3: Rewrite integral in terms of u

$\int 2u^3 du$

Step4: Integrate power function

$2 \cdot \frac{u^{4}}{4} = \frac{1}{2}u^4$

Step5: Substitute back to x

Replace $u$ with $\sin(x)$: $\frac{1}{2}\sin^4(x)$

Answer:

$\frac{1}{2}\sin^4(x)$