Solutions - 0040-01

(1) Notice odd power on $\cos x$. Swap the even bunch:

$${\sin }^{2}x\cdot {\cos }^{3}x\gg \gg {\sin }^{2}x(1-{\sin }^{2}x)\cos x$$

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(2) Integrate with $u$-sub setting $u=\sin x$ and thus $du=\cos xdx$:

$$\begin{array}{cc}\int {\sin }^{2}x(1-{\sin }^{2}x)\cos xdx& \gg \gg \int u^2(1-u^2)du\\ & \\ & \gg \gg \frac{u^3}{3}-\frac{u^5}{5}+C\\ & \\ & \gg \gg \frac{{\sin }^{3}x}{3}-\frac{{\sin }^{5}x}{5}+C\end{array}$$