Question

evaluate the indefinite integral. \\(\int 4 \sin^2(x) \cos(x) \\, dx = \square + c\\).

Explanation:

Step1: Use substitution

Let \(u = \sin(x)\), then \(du=\cos(x)dx\).
The integral \(\int4\sin^{2}(x)\cos(x)dx\) becomes \(\int4u^{2}du\).

Step2: Integrate with power rule

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(\int4u^{2}du\), we have \(4\times\frac{u^{3}}{3}+C=\frac{4}{3}u^{3}+C\).

Step3: Substitute back

Since \(u = \sin(x)\), the result is \(\frac{4}{3}\sin^{3}(x)+C\).

Answer:

\(\frac{4}{3}\sin^{3}(x)\)