Section 6-2 - Trig Integrals

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Chapter 6: Techniques of Integration

Section 6.2: Trigonometric Integrals

Essentials

Table 6.2.1 lists two reduction formulas for integrals of integer-powers of products of sines and cosines. These formulas can be established by parts integration followed by trigonometric and algebraic manipulations.

$\int \cos^{m} (x) \sin^{n} (x) &DifferentialD; x$ = $\frac{\cos^{m - 1} (x) \sin^{n + 1} (x)}{m + n} + \frac{m - 1}{m + n}$ $\int \cos^{m - 2} (x) \sin^{n} (x) &DifferentialD; x$

$n \geq 0$ $m \geq 2$

$\int \cos^{m} (x) \sin^{n} (x) &DifferentialD; x$ = $- \frac{\cos^{m + 1} (x) \sin^{n - 1} (x)}{m + n} + \frac{n - 1}{m + n}$ $\int \cos^{m} (x) \sin^{n - 2} (x) &DifferentialD; x$

$m \geq 0$

$n \geq 2$

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

If one of $m$ and $n$ is odd ( $2 k + 1$ ), the integral on the left in Table 6.2.1 can be evaluated by the alternative strategy detailed in Table 6.2.2.

$\int \cos^{m} (x) \sin^{2 k + 1} (x) &DifferentialD; x$	= $\int \cos^{m} (x) \sin^{2 k} (x) \sin (x) &DifferentialD; x$
	$= \int \cos^{m} (x) {(1 - \cos^{2} (x))}^{k} \sin (x) &DifferentialD; x$	${\begin{array}{c} u = \cos (x) \\ du = - \sin (x) dx \end{array}$
	$= - \int u^{m} {(1 - u^{2})}^{k} &DifferentialD; u$

$\int \cos^{2 k + 1} (x) \sin^{n} (x) &DifferentialD; x$	$= \int \cos^{2 k} (x) \cos (x) \sin^{n} (x) &DifferentialD; x$
	$= \int {(1 - \sin^{2} (x))}^{k} \cos (x) \sin^{n} (x) &DifferentialD; x$	${\begin{array}{c} u = \sin (x) \\ du = \cos (x) dx \end{array}$
	$= \int {(1 - u^{2})}^{k} u^{n} &DifferentialD; u$

Table 6.2.2 Alternate approach to the reduction formulas in Table 6.2.1

If $n = 0$ in the first formula of Table 6.2.1, or $m = 0$ in the second, the special cases listed in Table 6.2.3 result.

$\int \cos^{m} (x) &DifferentialD; x =$ $\frac{1}{m} \cos^{m - 1} (x) \sin (x) + \frac{m - 1}{m}$ $\int \cos^{m - 2} (x) &DifferentialD; x$

$\int \sin^{n} (x) &DifferentialD; x$ = $- \frac{1}{n} \cos (x) \sin^{n - 1} (x) + \frac{n - 1}{n}$ $\int \sin^{n - 2} (x) &DifferentialD; x$

Table 6.2.3 Special cases of the reduction formulas in Table 6.2.1

If $m = n = 2$ in Table 6.2.3, the special cases listed in Table 6.2.4 result.

$\int \cos^{2} (x) &DifferentialD; x$ = $\frac{1}{2} (\cos (x) \sin (x) + \int 1 &DifferentialD; x)$ = $\frac{1}{2} (x + \cos (x) \sin (x))$

$\int \sin^{2} (x) &DifferentialD; x$ = $\frac{1}{2} (- \cos (x) \sin (x) + \int 1 &DifferentialD; x)$ = $\frac{1}{2} (x - \cos (x) \sin (x))$

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

The alternative to the results in Table 6.2.4 is to remember (and apply) the trig identities embodied in Table 6.2.5, namely, $\cos^{2} (x) = (1 + \cos (2 x)) / 2, \sin^{2} (x) = (1 - \cos (2 x)) / 2$ , and $\sin (2 x) = 2 \sin (x) \cos (x)$ .

$\int \cos^{2} (x) &DifferentialD; x$	$= \int \frac{1}{2} (1 + \cos (2 x)) &DifferentialD; x$	$= \frac{1}{2} (x + \frac{1}{2} \sin (2 x))$	$= \frac{1}{2} (x + \sin (x) \cos (x))$
$\int \sin^{2} (x) &DifferentialD; x$	$= \int \frac{1}{2} (1 - \cos (2 x)) &DifferentialD; x$	$= \frac{1}{2} (x - \frac{1}{2} \sin (2 x))$	$= \frac{1}{2} (x - \sin (x) \cos (x))$
Table 6.2.5 The results in Table 6.2.4 established by application of trig identities

Table 6.2.6 lists integration formulas for products of sines and cosines whose arguments are integer multiples of $x$ .

$\int \sin (m x) \sin (n x) dx$	$= \frac{\sin ((m - n) x)}{2 (m - n)} - \frac{\sin ((m + n) x)}{2 (m + n)}$
$\int \cos (m x) \cos (n x) dx$	$= \frac{\sin ((m - n) x)}{2 (m - n)} + \frac{\sin ((m + n) x)}{2 (m + n)}$
$\int \sin (m x) \cos (n x) dx$	$= - \frac{\cos ((m - n) x)}{2 (m - n)} - \frac{\cos ((m + n) x)}{2 (m + n)}$
Table 6.2.6 Integral formulas for products of sines and cosines

The formulas in Table 6.2.6 follow from the application of the basic trig identities listed in Table 6.2.7.

$\sin (x) \sin (y)$	$= \frac{1}{2} (\cos (x - y) - \cos (x + y))$
$\cos (x) \cos (y)$	$= \frac{1}{2} (\cos (x - y) + \cos (x + y))$
$\sin (x) \cos (y)$	$= \frac{1}{2} (\sin (x - y) + \sin (x + y))$
Table 6.2.7 Trig identities for the integrals in Table 6.2.6

Integrals of the form $\int \tan^{m} (x) \sec^{n} (x) &DifferentialD; x$ , where either $n$ is even ( $2 k)$ or $m$ is odd ( $2 k + 1$ ), yield to a strategy similar to that in Table 6.2.2. Table 6.2.8 lists these results.

$\int \tan^{m} (x) \sec^{2 k} (x) &DifferentialD; x$	$= \int \tan^{m} (x) {(\sec^{2} (x))}^{k - 1} \sec^{2} (x) &DifferentialD; x$
	$= \int \tan^{m} (x) {(1 + \tan^{2} (x))}^{k - 1} \sec^{2} (x) &DifferentialD; x$	${\begin{array}{c} u = \tan (x) \\ du = \sec^{2} (x) dx \end{array}$
	$= \int u^{m} {(1 + u^{2})}^{k - 1} du$

$\int \tan^{2 k + 1} (x) \sec^{n} (x) &DifferentialD; x$	$= \int {(\tan^{2} (x))}^{k} \sec^{n - 1} (x) \sec (x) \tan (x) &DifferentialD; x$
	$= \int {(\sec^{2} (x) - 1)}^{k} \sec^{n - 1} (x) \sec (x) \tan (x) &DifferentialD; x$	${\begin{array}{c} u = \sec (x) \\ du = \sec (x) \tan (x) dx \end{array}$
	$= \int {(u^{2} - 1)}^{k} u^{n - 1} &DifferentialD; u$

Table 6.2.8 Special cases of the integral $\int \tan^{m} (x) \sec^{n} (x) &DifferentialD; x$

Note the related reduction formula

$\int \sec^{2 k + 1} (x) dx = \frac{1}{2 k} (\sec^{2 k - 1} (x) \tan (x) + (2 k - 1) \int \sec^{2 k - 1} (x) dx)$

that is derived in Example 6.2.9.

Integrals of the form $\int \cot^{m} (x) \csc^{n} (x) dx$ , where either $n$ is even ( $2 k)$ or $m$ is odd ( $2 k + 1$ ), yield to a strategy similar to that in Table 6.2.8. Table 6.2.9 lists these results.

$\int \cot^{m} (x) \csc^{2 k} (x) dx$	$= \int \cot^{m} (x) {(\csc^{2} (x))}^{k - 1} \csc^{2} (x) dx$
	$= \int \cot^{m} (x) {(1 + \cot^{2} (x))}^{k - 1} \csc^{2} (x) dx$	${\begin{array}{c} u = \cot (x) \\ du = \csc^{2} (x) dx \end{array}$
	$= \int u^{m} {(1 + u^{2})}^{k - 1} du$

$\int \cot^{2 k + 1} (x) \csc^{n} (x) dx$	$= \int {(\cot^{2} (x))}^{k} \csc^{n - 1} (x) \csc (x) \cot (x) dx$
	$= \int {(\csc^{2} (x) - 1)}^{k} \csc^{n - 1} (x) \csc (x) \cot (x) dx$	${\begin{array}{c} u = \csc (x) \\ du = - \csc (x) \cot (x) dx \end{array}$
	$= \int {(u^{2} - 1)}^{k} u^{n - 1} du$

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

Table 3.10.1 lists antiderivatives for $\tan (x)$ and $\cot (x)$ ; Table 6.2.10 lists and deduces antiderivatives for $\sec (x)$ and $\csc (x)$ .

$\int \sec (x) dx$	$= \int \sec (x) \frac{\sec (x) + \tan (x)}{\sec (x) + \tan (x)} dx$
	$= \int \frac{\sec^{2} (x) + \sec (x) \tan (x)}{\sec (x) + \tan (x)} dx$	${\begin{array}{c} u = \sec (x) + \tan (x) \\ du = (\sec (x) \tan (x) + \sec^{2} (x)) dx \end{array}$
	$= \int \frac{du}{u}$
	$= \ln (\|u\|)$
	$= \ln (\|\sec (x) + \tan (x)\|)$
$\int \csc (x) &DifferentialD; x$	$= \int \csc (x) \frac{\csc (x) + \cot (x)}{\csc (x) + \cos (x)} &DifferentialD; x$
	$= \int \frac{\csc^{2} (x) + \csc (x) \cot (x)}{\csc (x) + \cot (x)} &DifferentialD; x$	${\begin{array}{c} u = \csc (x) + \cot (x) \\ du = - (\csc (x) \cot (x) + \csc^{2} (x)) dx \end{array}$
	$= - \int \frac{du}{u}$
	$= - \ln (\|u\|)$
	$= - \ln (\|\csc (x) + \cot (x)\|)$ = $\ln (\|\csc (x) - \cot (x)\|)$

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

Examples

Example 6.2.1	Evaluate the indefinite integral $\int \cos^{3} (x) \sin^{2} (x) &DifferentialD; x$ .
Example 6.2.2	Evaluate the indefinite integral $\int \cos^{4} (x) &DifferentialD; x$ .
Example 6.2.3	Evaluate the indefinite integral $\int \tan^{4} (x) \sec^{6} (x) &DifferentialD; x$ .
Example 6.2.4	Evaluate the indefinite integral $\int \tan^{3} (x) &DifferentialD; x$ .
Example 6.2.5	Evaluate the indefinite integral $\int \sec^{3} (x) &DifferentialD; x$ .
Example 6.2.6	Evaluate the indefinite integral $\int \sin (7 x) \sin (6 x) &DifferentialD; x$ .
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 $\int \sec^{2 k + 1} (x) dx = \frac{1}{2 k} (\sec^{2 k - 1} (x) \tan (x) + (2 k - 1) \int \sec^{2 k - 1} (x) dx)$ .
Example 6.2.10	Evaluate the indefinite integral $\int \csc^{3} (x) &DifferentialD; x$ .

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