Integration using Euler's formula


In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely and and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts, and is sufficiently powerful to integrate any rational expression involving trigonometric functions.

Euler's formula

Euler's formula states that
Substituting for gives the equation
because cosine is an even function and sine is odd. These two equations can be solved for the sine and cosine to give

Examples

First example

Consider the integral
The standard approach to this integral is to use a half-angle formula to simplify the integrand. We can use Euler's identity instead:
At this point, it would be possible to change back to real numbers using the formula. Alternatively, we can integrate the complex exponentials and not change back to trigonometric functions until the end:

Second example

Consider the integral
This integral would be extremely tedious to solve using trigonometric identities, but using Euler's identity makes it relatively painless:
At this point we can either integrate directly, or we can first change the integrand to and continue from there.
Either method gives

Using real parts

In addition to Euler's identity, it can be helpful to make judicious use of the real parts of complex expressions. For example, consider the integral
Since is the real part of, we know that
The integral on the right is easy to evaluate:
Thus:

Fractions

In general, this technique may be used to evaluate any fractions involving trigonometric functions. For example, consider the integral
Using Euler's identity, this integral becomes
If we now make the substitution, the result is the integral of a rational function:
One may proceed using partial fraction decomposition.