Trigonometric Integrals

When the integrand is primarily or exclusively based on trigonometric functions, the following techniques are useful.

Powers of Sine and Cosine

We will give a general method to solve generally integrands of the form $\cos ^{m}(x)\cdot \sin ^{n}(x)$ . First let us work through an example.

$\int \cos ^{3}(x)\sin ^{2}(x)dx$

Notice that the integrand contains an odd power of cos. So rewrite it as

$\int \cos ^{2}(x)\sin ^{2}(x)\cos(x)dx$

We can solve this by making the substitution $u=\sin(x)$ so $du=\cos(x)dx$ . Then we can write the whole integrand in terms of $u$ by using the identity

$\cos ^{2}(x)=1-\sin ^{2}(x)=1-u^{2}$ .

So

$\int \cos ^{3}(x)\sin ^{2}(x)dx$	$=\int \cos ^{2}(x)\sin ^{2}(x)\cos(x)dx$
	$=\int (1-u^{2})u^{2}du$
	$=\int u^{2}du-\int u^{4}du$
	$={\frac {u^{3}}{3}}+{\frac {u^{5}}{5}}+C$
	$={\frac {\sin ^{3}(x)}{3}}-{\frac {\sin ^{5}(x)}{5}}+C$

This method works whenever there is an odd power of sine or cosine.

To evaluate $\int \cos ^{m}(x)\sin ^{n}(x)dx$ when either $m$ or $n$ is odd.

• If $m$ is odd substitute $u=\sin(x)$ and use the identity $\cos ^{2}(x)=1-\sin ^{2}(x)=1-u^{2}$ .
• If $n$ is odd substitute $u=\cos(x)$ and use the identity $\sin ^{2}(x)=1-\cos ^{2}(x)=1-u^{2}$ .

Example

Find $\int \limits _{0}^{\frac {\pi }{2}}\cos ^{40}(x)\sin ^{3}(x)dx$ .

As there is an odd power of $\sin$ we let $u=\cos(x)$ so $du=-\sin(x)dx$ . Notice that when $x=0$ we have $u=\cos(0)=1$ and when $x={\frac {\pi }{2}}$ we have $u=\cos({\tfrac {\pi }{2}})=0$ .

$\int \limits _{0}^{\frac {\pi }{2}}\cos ^{40}(x)\sin ^{3}(x)dx$	$=\int \limits _{0}^{\frac {\pi }{2}}\cos ^{40}(x)\sin ^{2}(x)\sin(x)dx$
	$=-\int \limits _{1}^{0}u^{40}(1-u^{2})du$
	$=\int \limits _{0}^{1}u^{40}(2-u^{2})du$
	$=\int \limits _{0}^{5}(u^{40}-u^{42})du$
	$=\left({\frac {u^{41}}{41}}-{\frac {u^{43}}{43}}\right){\Bigg |}_{0}^{1}$
	$={\frac {1}{41}}-{\frac {1}{43}}$

When both $m$ and $n$ are even, things get a little more complicated.

To evaluate $\int \cos ^{m}(x)\sin ^{n}(x)dx$ when both $m$ and $n$ are even.

Use the identities $\sin ^{2}(x)={\frac {1-\cos(2x)}{2}}$ and $\cos ^{2}(x)={\frac {1+\cos(2x)}{2}}$ .

Example

Find $\int \sin ^{2}(x)\cos ^{4}(x)dx$ .

As $\sin ^{2}(x)={\frac {1-\cos(2x)}{2}}$ and $\cos ^{2}(x)={\frac {1+\cos(2x)}{2}}$ we have

$\int \sin ^{2}(x)\cos ^{4}(x)dx=\int \left({\frac {1-\cos(2x)}{2}}\right)\left({\frac {1+\cos(2x)}{2}}\right)^{2}dx$

and expanding, the integrand becomes

${\frac {1}{8}}\int \left(1-\cos ^{2}(2x)+\cos(2x)-\cos ^{3}(2x)\right)dx$

Using the multiple angle identities

$I$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{8}\left(\int 1dx-\int\cos^2(2x)dx+\int\cos(2x)dx-\int\cos^3(2x)dx\right)}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{8}\left(x-\frac{1}{2}\int\Big(1+\cos(4x)\Big)dx+\frac{\sin(2x)}{2}-\int\cos^2(2x)\cos(2x)dx\right)}
	TODO: CORRECT FORMULA$={\frac {1}{164}}\left(x+\sin(2x)+\int \cos(4x)dx-2\int {\Big (}1-\sin ^{2}(2x){\Big )}\cos(2x)dx\right)$

then we obtain on evaluating

$I={\frac {x}{16}}-{\frac {\sin(4x)}{64}}+{\frac {\sin ^{3}(2x)}{48}}+C$

Powers of Tan and Secant

To evaluate $\int \tan ^{m}(x)\sec ^{n}(x)dx$ .

1. If $n$ is even and $n\geq 2$ then substitute $u=tan(x)$ and use the identity $\sec ^{2}(x)=1+\tan ^{2}(x)$ .
2. If $n$ and $m$ are both odd then substitute $u=\sec(x)$ and use the identity $\tan ^{2}(x)=\sec ^{2}(x)-1$ .
3. If $n$ is odd and $m$ is even then use the identity $\tan ^{2}(x)=\sec ^{2}(x)-1$ and apply a reduction formula to integrate $\sec ^{j}(x)dx$ , using the examples below to integrate when $j=1,2$ .

Example 1

Find $\int \sec ^{2}(x)dx$ .

There is an even power of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(x)} . Substituting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\tan(x)} gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=\sec^2(x)dx} so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sec^2(x)dx=\int du=u+C=\tan(x)+C.}

Example 2

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\tan(x)dx} .

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\cos(x)} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=-\sin(x)dx} . Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\tan(x)dx}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\frac{\sin(x)}{\cos(x)}dx}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int -\frac{du}{u}}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-\ln|u|+C}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =-\ln\Big|\cos(x)\Big|+C}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\ln\Big|\sec(x)\Big|+C}

Example 3

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sec(x)dx} .

The trick to do this is to multiply and divide by the same thing like this:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sec(x)dx}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\sec(x)\frac{\sec(x)+\tan(x)}{\sec(x)+\tan(x)}dx}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\frac{\sec^2(x)+\sec(x)\tan(x)}{\sec(x)+\tan(x)}dx}

Making the substitution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=\sec(x)+\tan(x)} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=\sec(x)\tan(x)+\sec^2(x)dx} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sec(x)dx}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\frac{du}{u}}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\ln|u|+C}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln\Big|\sec(x)+\tan(x)\Big|+C}

More trigonometric combinations

For the integrals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(nx)\cos(mx)dx} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(nx)\sin(mx)dx} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\cos(nx)\cos(mx)dx} use the identities

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(a)\cos(b)=\frac{\sin(a+b)+\sin(a-b)}{2}}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(a)\sin(b)=\frac{\cos(a-b)-\cos(a+b)}{2}}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(a)\cos(b)=\frac{\cos(a-b)+\cos(a+b)}{2}}

Example 1

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(3x)\cos(5x)dx} .

We can use the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(a)\cos(b)=\frac{\sin(a+b)+\sin(a-b)}{2}} , so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(3x)\cos(5x)=\frac{\sin(8x)+\sin(-2x)}{2}}

Now use the oddness property of $\sin(x)$ to simplify

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(3x)\cos(5x)=\frac{\sin(8x)-\sin(2x)}{2}}

And now we can integrate

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(3x)\cos(5x)dx}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\Big(\frac{\sin(8x)-\sin(2x)}{2}\Big)dx}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{\cos(2x)}{4}-\frac{\cos(8x)}{16}+C}

Example 2

Find:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(x)\sin(2x)dx} .

Using the identities

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(x)\sin(2x)=\frac{\cos(-x)-\cos(3x)}{2}=\frac{\cos(x)-\cos(3x)}{2}}

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\sin(x)\sin(2x)dx}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{1}{2}\int\Big(\cos(x)-\cos(3x)\Big)dx}
	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{\sin(x)}{2}-\frac{\sin(3x)}{6}+C}

Resources

• Trigonometric Integrals Involving Powers of Sine and Cosine - Part 1 by James Sousa

• Trigonometric Integrals Involving Powers of Sine and Cosine - Part 2 by James Sousa

• Trigonometric Integrals Involving Powers of Secant and Tangent - Part 1 by James Sousa

• Trigonometric Integrals Involving Powers of Secant and Tangent - Part 1 by James Sousa

• Trigonometric Integrals - Part 1 of 6 by patrickJMT

• Trigonometric Integrals - Part 2 of 6 by patrickJMT

• Trigonometric Integrals - Part 3 of 6 by patrickJMT

• Trigonometric Integrals - Part 4 of 6 by patrickJMT

• Trigonometric Integrals - Part 5 of 6 by patrickJMT

• Trigonometric Integrals - Part 6 of 6 by patrickJMT

• Trigonometric integrals - sin^mcos^n, odd m by Kriata King

• Trigonometric integrals - sin^mcos^n, odd n by Kriata King

• Trigonometric integrals - sin^mcos^n, m and n even by Kriata King

• Integrals of trigonometric functions, tan^msec^n, even n by Krista King

• Integrals of trigonometric functions, tan^msec^n, odd m by Krista King

Licensing

Content obtained and/or adapted from:

• Trigonometric integrals, Wikibooks: Calculus/Integration techniques under a CC BY-SA license