Solutions - 0040-04

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

$$\begin{array}{c}{\cos }^{7}x\gg \gg {({\cos }^{2}x)}^3\cos x\\ \\ \gg \gg {(1-{\sin }^{2}x)}^3\cos x\end{array}$$

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

$$\begin{array}{c}\int {\cos }^{7}xdx\gg \gg \int {(1-{\sin }^{2}x)}^3\cos xdx\\ \\ \gg \gg \int {(1-u^2)}^{3}du\gg \gg \int 1-3u^2+3u^4-u^{6}du\\ \\ \gg \gg u-u^3+3\frac{u^5}{5}-\frac{u^7}{7}+C\end{array}$$

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(3) Convert back to $x$:

$$\gg \gg \sin x-{\sin }^{3}x+\frac{3}{5}{\sin }^{5}x-\frac{1}{7}{\sin }^{7}x+C$$