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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