$\int {\cos }^m\left(x\right){\sin }^n\left(x\right) \mathit{\ⅆ}x$ = $\frac{{\cos }^{m-1}\left(x\right){\sin }^{n\+1}\left(x\right)}{m\+n}\+\frac{m-1}{m\+n}$ $\int {\cos }^{m-2}\left(x\right){\sin }^n\left(x\right) \mathit{\ⅆ}x$ 

$n\ge 0$ $m\ge 2$

$\int {\cos }^m\left(x\right){\sin }^n\left(x\right) \mathit{\ⅆ}x$ = $-\frac{{\cos }^{m\+1}\left(x\right){\sin }^{n-1}\left(x\right)}{m\+n}\+\frac{n-1}{m\+n}$ $\int {\cos }^m\left(x\right){\sin }^{n-2}\left(x\right) \mathit{\ⅆ}x$ 

$m\ge 0$

$n\ge 2$

Table 6.2.1   Reduction formulas for integrals of products of powers of sines and cosines

Table 6.2.2   Alternate approach to the reduction formulas in Table 6.2.1

$\int {\cos }^m\left(x\right) \mathit{\ⅆ}x\=$$\frac{1}{m}{\cos }^{m-1}\left(x\right)\sin \left(x\right)\+\frac{m-1}{m}$ $\int {\cos }^{m-2}\left(x\right) \mathit{\ⅆ}x$ $$

$\int {\sin }^n\left(x\right) \mathit{\ⅆ}x$ = $-\frac{1}{n}\cos \left(x\right){\sin }^{n-1}\left(x\right)\+\frac{n-1}{n}$ $\int {\sin }^{n-2}\left(x\right) \mathit{\ⅆ}x$ 

Table 6.2.3   Special cases of the reduction formulas in Table 6.2.1

$\int {\cos }^2\left(x\right) \mathit{\ⅆ}x$ = $\frac{1}{2}\left(\cos \left(x\right)\sin \left(x\right)\+\int 1 \mathit{\ⅆ}x\right)$ = $\frac{1}{2}\left(x\+\cos \left(x\right)\sin \left(x\right)\right)$

$\int {\sin }^2\left(x\right) \mathit{\ⅆ}x$ = $\frac{1}{2}\left(-\cos \left(x\right)\sin \left(x\right)\+\int 1 \mathit{\ⅆ}x\right)$ = $\frac{1}{2}\left(x-\cos \left(x\right)\sin \left(x\right)\right)$

Table 6.2.4   Special cases $m\=n\=2$ in Table 6.2.3

$\int {\cos }^2\left(x\right) \mathit{\ⅆ}x$

$\= \int \frac{1}{2}\left(1\+\cos \left(2 x\right)\right) \mathit{\ⅆ}x$

$\=\frac{1}{2}\left(x\+\frac{1}{2}\sin \left(2 x\right)\right)$

$\=\frac{1}{2}\left(x\+\sin \left(x\right)\cos \left(x\right)\right)$

$\int {\sin }^2\left(x\right) \mathit{\ⅆ}x$

$\= \int \frac{1}{2}\left(1-\cos \left(2 x\right)\right) \mathit{\ⅆ}x$

$\=\frac{1}{2}\left(x-\frac{1}{2}\sin \left(2 x\right)\right)$

$\=\frac{1}{2}\left(x-\sin \left(x\right)\cos \left(x\right)\right)$

Table 6.2.5   The results in Table 6.2.4 established by application of trig identities

$\int \sin \left(m x\right)\sin \left(n x\right) \mathrm{dx}$

$\=\frac{\sin \left(\left(m-n\right) x\right)}{2\left(m-n\right)}-\frac{\sin \left(\left(m\+n\right) x\right)}{2\left(m\+n\right)}$

$\int \cos \left(m x\right)\cos \left(n x\right) \mathrm{dx}$

$\=\frac{\sin \left(\left(m-n\right) x\right)}{2\left(m-n\right)}\+\frac{\sin \left(\left(m\+n\right) x\right)}{2\left(m\+n\right)}$

$\int \sin \left(m x\right)\cos \left(n x\right) \mathrm{dx}$

$\=-\frac{\cos \left(\left(m-n\right) x\right)}{2\left(m-n\right)}-\frac{\cos \left(\left(m\+n\right) x\right)}{2\left(m\+n\right)}$

Table 6.2.6   Integral formulas for products of sines and cosines

$\sin \left(x\right)\sin \left(y\right)$

$\=\frac{1}{2}\left(\cos \left(x-y\right)-\cos \left(x\+y\right)\right)$

$\cos \left(x\right)\cos \left(y\right)$

$\=\frac{1}{2}\left(\cos \left(x-y\right)\+\cos \left(x\+y\right)\right)$

$\sin \left(x\right)\cos \left(y\right)$

$\=\frac{1}{2}\left(\sin \left(x-y\right)\+\sin \left(x\+y\right)\right)$

Table 6.2.7   Trig identities for the integrals in Table 6.2.6

Table 6.2.8   Special cases of the integral $\int {\tan }^m\left(x\right){\sec }^n\left(x\right) \mathit{\ⅆ}x$

Table 6.2.9   Special cases of the integral $\int {\cot }^m\left(x\right){\csc }^n\left(x\right) \mathrm{dx}$

Table 6.2.10   Antiderivatives for $\sec \left(x\right)$ and $\csc \left(x\right)$ 

Example 6.2.1

Evaluate the indefinite integral $\int {\cos }^3\left(x\right){\sin }^2\left(x\right) \mathit{\ⅆ}x$.

Example 6.2.2

Evaluate the indefinite integral $\int {\cos }^4\left(x\right) \mathit{\ⅆ}x$.

Example 6.2.3

Evaluate the indefinite integral $\int {\tan }^4\left(x\right){\sec }^6\left(x\right) \mathit{\ⅆ}x$.

Example 6.2.4

Evaluate the indefinite integral $\int {\tan }^3\left(x\right) \mathit{\ⅆ}x$.

Example 6.2.5

Evaluate the indefinite integral $\int {\sec }^3\left(x\right) \mathit{\ⅆ}x$.

Example 6.2.6

Evaluate the indefinite integral $\int \sin \left(7 x\right)\sin \left(6 x\right) \mathit{\ⅆ}x$.

Example 6.2.7

Derive the first reduction formula in Table 6.2.1.

Example 6.2.8

Derive the second reduction formula in Table 6.2.1.

Example 6.2.9

Derive the reduction formula $\int {\sec }^{2 k\+1}\left(x\right) \mathrm{dx}\=\frac{1}{2 k}\left({\sec }^{2 k-1}\left(x\right)\tan \left(x\right)\+\left(2 k-1\right)\int {\sec }^{2 k-1}\left(x\right) \mathrm{dx}\right)$.

Example 6.2.10

Evaluate the indefinite integral $\int {\csc }^3\left(x\right) \mathit{\ⅆ}x$.