# Integration of Trigonometric Functions

## Integrals of the form $$\displaystyle \int \sin^m x\, \cos ^n x\,dx$$

The integrals of the form $$\displaystyle \int \sin^m x\, \cos ^n x\,dx$$ are very easy to evaluate in the case when one (or both) of the numbers $$m$$ and $$n$$ is odd integer.

• If $$m$$ is odd, assume that $$m=2k+1$$. Then the integral can be rewritten as $\int \sin^m x\, \cos ^nx\,dx= \int \sin^{2k}x\, \cos^nx\cdot\sin x\,dx.$ The substitution $$u=\cos x$$ has $$du=-\sin x\,dx$$ and transforms the integral into $\int \sin^m x\, \cos ^nx\,dx= -\int (1-u^2)^k u^n\,du$ which is easy to evaluate. If $$k\geq 0$$ then $$(1-u^2)^k$$ can be expanded and the integrand becomes a sum of the functions of the form $$u^{\alpha}$$ (where $$\alpha\in\mathbb R$$). If $$k> 0$$ and if $$n$$ is an integer, then we are dealing with a rational function for which we have a method for integration.

• If $$n$$ is odd, we may assume that $$n=2l+1$$ for some integer $$l$$. The integral can be rewritten as $\int \sin^mx\, \cos^nx\,dx=\int\sin^mx \, \cos^{2l}x \cdot\cos x\,dx.$ The substitution $$u=\sin x$$ has $$du=\cos x\,dx$$ and simplifies the integral in the following way: $\int \sin^mx\, \cos^nx\,dx=\int u^m(1-u^2)^l\,du.$ Similarly as in the above paragraph we can solve this integral in the case when $$l\geq 0$$ and $$m$$ is an arbitrary real number, or when $$l< 0$$ and $$m$$ is an integer.

Example 1

Evaluate the integral $$\displaystyle \int \sqrt[3]{\cos x}\cdot \sin^3x\,dx$$.

Example 2

Evaluate the integral $$\displaystyle \int \sec x\,dx=\int\frac1{\cos x}\,dx$$.

If both $$m$$ and $$n$$ are even, then use the formulas $$\cos^2\theta=\frac{1+\cos(2\theta)}2$$ and $$\sin^2\theta=\frac{1-\cos(2\theta)}2$$ to reduce them to the previous case.

Example 3

Evaluate the integral $$\displaystyle \int \cos^4x\,\sin^2 x\,dx$$.

## Integrals of the form $$\displaystyle \int \frac{P(\tan x)}{Q(\tan x)}\,dx$$

If the integrand is a rational function of $$\tan x$$ the substitution $$\tan x=t$$ transforms the integral in a rational function because $$x=\tan^{-1}t$$ and consequently $$dx=\frac1{1+t^2}\,dt$$ which is rational function as well.

It is worth noting that many integrals that do not appear to be of this form can be re-written as rational functions of $$\tan$$ by dividing both numerator and denominator by the appropriate factor.

Example 3

Evaluate the integral $$\displaystyle \int \frac{2\sin^3x+\sin^2x\cos x-4\sin x\cos^2x+3\cos^3x}{\left(\sin^2x-\sin x\cos x-2\cos^2x\right)\cos x}\,dx$$.

## Integrals of the form $$\displaystyle \int\frac{P(\sin x, \cos x)}{Q(\sin x, \cos x)}\,dx$$

The magical substitution $$u=\tan\frac{x}2$$ can bring the most general trigonometric integral to an integral of a rational function.

The the substitution owes its power to the fact that $$dx$$, $$\sin x$$, and $$\cos x$$ can be expressed in terms of $$u$$ in a relatively simple manner. First of all, $$x=2\tan^{-1}u$$ and $$dx=\frac{2du}{u^2+1}$$. We need to express $$\sin x$$ and $$\cos x$$ in terms of $$u$$. \begin{eqnarray*}\sin x&=&2\sin\frac{x}2\cos\frac{x}2=2\frac{\sin{x}2}{\cos\frac{x}2}\cdot\cos^2\frac{x}2=2\tan\frac{x}2\cdot \frac1{\frac1{\cos^2\frac{x}2}}=\frac{2u}{1+u^2}\\ \cos x&=&\cos^2\frac{x}2-\sin^2\frac{x}2=\cos^2\frac{x}2\left(1-\tan\frac{x}2\right)=\frac1{\frac1{\cos^2\frac{x}2}}\left(1-u^2\right)=\frac{1-u^2}{1+u^2}.\end{eqnarray*}

Let us summarize the substitution: \begin{eqnarray*} u&=&\tan\frac{x}2\\ dx&=&\frac{2\,du}{u^2+1}\\ \sin x&=&\frac{2u}{u^2+1}\\ \cos x&=&\frac{1-u^2}{1+u^2}.\end{eqnarray*}

Example 4

Evaluate the integral $$\displaystyle \int\frac{1}{1+\sin x}\,dx$$.

In the beginning we learned how to use the substitutions $$u=\sin x$$, $$u=\cos x$$, and $$u=\tan x$$. This paragraph showed that these substitutions are unnecessary, as the substitution $$u=\tan\frac{x}2$$ is more powerful than any of the substitutions we covered. However, the integrals resulting from substituting $$u=\tan\frac{x}2$$ may be very unpleasant. One should attempt to use $$u=\sin x$$, $$u=\cos x$$, or $$u=\tan x$$ whenever possible.

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