Solve each of the integrals in Exercises 21–66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions.)

$\int {\sin }^{13}\left(x\right){\cos }^5\left(x\right)dx$

The given integral is$\int {\sin }^{13}\left(x\right){\cos }^5\left(x\right)dx$.

• Use the trigonometric identities to rewrite the integral as follows:

$\int {\sin }^{13}\left(x\right){\cos }^5\left(x\right)dx=\int {\left(1-{\cos }^2\left(x\right)\right)}^6\sin \left(x\right){\cos }^5\left(x\right)dx$

• Consider role="math" localid="1649012126134" $\cos x=u$, so $-\sin xdx=du$.
• Substitute and simplify the integral.

$\begin{array}{rcl}\int {\left(1-{\cos }^2\left(x\right)\right)}^6\sin \left(x\right){\cos }^5\left(x\right)dx& =& \int \hspace{0.17em}-u^5{\left(1-u^2\right)}^{6}du\\ & =& -\int \left(1-6u^2+15u^4-20u^6+15u^8-6u^{10}+u^{12}\right)u^{5}du\\ & =& -\int \left(u^5-6u^7+15u^9-20u^{11}+15u^{13}-6u^{15}+u^{17}\right)du\end{array}$

• Apply the sum rule on the obtained integral.

$=-\int \left(u^5-6u^7+15u^9-20u^{11}+15u^{13}-6u^{15}+u^{17}\right)du==-\int \hspace{0.17em}{u}^{5}du+\int \hspace{0.17em}6u^{7}du-\int \hspace{0.17em}15u^{9}du+\int \hspace{0.17em}20u^{11}du-\int \hspace{0.17em}15u^{13}du+\int \hspace{0.17em}6u^{15}du-\int \hspace{0.17em}{u}^{17}du$

• Perform integration on the obtained integral.

$-\int \hspace{0.17em}{u}^{5}du+\int \hspace{0.17em}6u^{7}du-\int \hspace{0.17em}15u^{9}du+\int \hspace{0.17em}20u^{11}du-\int \hspace{0.17em}15u^{13}du+\int \hspace{0.17em}6u^{15}du-\int \hspace{0.17em}{u}^{17}du=-\frac{u^6}{6}+\frac{3u^8}{4}-\frac{3u^{10}}{2}+\frac{5u^{12}}{3}-\frac{15u^{14}}{14}+\frac{3u^{16}}{8}-\frac{u^{18}}{18}$

• Substitute $u=\cos x$into the obtained integral to find the solution.
• So, the value of the integral is$-\frac{{\cos }^6\left(x\right)}{6}+\frac{3{\cos }^8\left(x\right)}{4}-\frac{3{\cos }^{10}\left(x\right)}{2}+\frac{5{\cos }^{12}\left(x\right)}{3}-\frac{15{\cos }^{14}\left(x\right)}{14}+\frac{3{\cos }^{16}\left(x\right)}{8}-\frac{{\cos }^{18}\left(x\right)}{18}$