Trigonometric Integrals

Trigonometric Integrals

The trigonometric integrals are a family of integrals which involve trigonometric functions ($\sin$, $\cos$, $\tan$, $\csc$, $\cot$, $\sec$). The following is a list of integrals of trigonometric functions. Some of them were computed using properties of the trigonometric functions, while others used techniques such as integration by parts. 

Generally, if the function, $\sin(x)$, is any trigonometric function, and $\cos(x)$ is its derivative, then

$\displaystyle{\int a\cos nx\;\mathrm{d}x = \frac{a}{n}\sin nx+C}$

In all formulas, the constant $a$ is assumed to be nonzero, while $C$ denotes the integration constant.

Integrands Involving Only Sine:

$\displaystyle{\int\sin ax\;\mathrm{d}x = -\frac{1}{a}\cos ax+C\,\! \\ \int\sin^2 {ax}\;\mathrm{d}x = \frac{x}{2} - \frac{1}{4a} \sin 2ax +C= \frac{x}{2} - \frac{1}{2a} \sin ax\cos ax +C\! \\ \int\sin^3 {ax}\;\mathrm{d}x = \frac{\cos 3ax}{12a} - \frac{3 \cos ax}{4a} +C\! \\ \int x\sin^2 {ax}\;\mathrm{d}x = \frac{x^2}{4} - \frac{x}{4a} \sin 2ax - \frac{1}{8a^2} \cos 2ax +C\! \\ \int x^2\sin^2 {ax}\;\mathrm{d}x = \frac{x^3}{6} - \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax - \frac{x}{4a^2} \cos 2ax +C\!}$

Integrands Involving Only Cosine:

$\int\cos ax\;\mathrm{d}x = \frac{1}{a}\sin ax+C$

$\int\cos^2 {ax}\;\mathrm{d}x = \frac{x}{2} + \frac{1}{4a} \sin 2ax +C = \frac{x}{2} + \frac{1}{2a} \sin ax\cos ax +C$

$\int\cos^n ax\;\mathrm{d}x = \frac{\cos^{n-1} ax\sin ax}{na} + \frac{n-1}{n}\int\cos^{n-2} ax\;\mathrm{d}x \qquad\mbox{(for }n>0\mbox{)}$

$\int x\cos ax\;\mathrm{d}x = \frac{\cos ax}{a^2} + \frac{x\sin ax}{a}+C$

$\int x^2\cos^2 {ax}\;\mathrm{d}x = \frac{x^3}{6} + \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax + \frac{x}{4a^2} \cos 2ax +C$

Integrands Involving Only Tangent:

$\begin{aligned}\int\tan ax\;\mathrm{d}x &= -\frac{1}{a}\ln \left|\cos ax \right|+C\\ & = \frac{1}{a}\ln \left|\sec ax \right|+C\end{aligned}$

$\displaystyle{\int\tan^n ax\;\mathrm{d}x = \frac{1}{a(n-1)}\tan^{n-1} ax-\int\tan^{n-2} ax\;\mathrm{d}x}$

where $n \neq 1$; and

$\displaystyle{\int\frac{\mathrm{d}x}{q \tan ax + p} = \frac{1}{p^2 + q^2}(px + \frac{q}{a}\ln \left|q\sin ax + p\cos ax \right|)+C }$

where $p^2 + q^2 \neq 0$.

Integrands Involving Only Secant:

$\displaystyle{\int \sec{ax} \, \mathrm{d}x = \frac{1}{a}\ln{\left| \sec{ax} + \tan{ax}\right|}+C}$

$\displaystyle{\int \sec^2{x} \, \mathrm{d}x = \tan{x}+C}$

Integrands involving only cosecant:

$\displaystyle{\int \csc{ax} \, \mathrm{d}x = \frac{1}{a}\ln{\left| \csc{ax}-\cot{ax}\right|}+C}$

$\displaystyle{\int \csc^2{x} \, \mathrm{d}x = -\cot{x}+C}$

Complicated Trigonometric Integrals

We now look at integrals involving the product of a power of $\sin x$ and a power of $\cos x$. Two simple examples of such integrals are $\int \sin^k x \cos x \; \mathrm d x$ and $\int \cos^k x \sin x\; \mathrm d x$ , which can be solved used the substitutions $u = \sin x$ and $u = \cos x$ , respectively. We now consider the more general case of $\int \sin^n x \cos^m x\; \mathrm d x$ , where $n$ and $m$ are positive integers.

• If $n$ is odd, we can pull out one factor of $\sin x$ , convert the rest to cosines using the identity $\sin^2 x + \cos^2 x = 1$ , and then use the substitution $u = \cos x$ .
• If $m$ is odd, we can pull out one factor of $\cos x$ , convert the rest to sines using the identity $\sin^2 x + \cos^2 x = 1$ , and then use the substitution $u = \sin x$.
• If both $n$ and $m$ are odd, we can use either of the above two methods.
• If both $n$ and $m$ are even, then we can try to use a combination of the following three identities: $\cos^2 x = \frac{1}{2} (1 + \cos 2x)$, $\sin^2 x = \frac{1}{2} (1 - \sin 2x)$, and $\sin x \cos x = \frac{1}{2} \sin 2x$.