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 ${\displaystyle e^{ix}}$ and ${\displaystyle e^{-ix}}$ 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 states that

Substituting ${\displaystyle -x}$ for ${\displaystyle x}$ 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

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 e^2ix + e^−2ix = 2 cos 2x. Alternatively, we can integrate the complex exponentials and not change back to trigonometric functions until the end:

Consider the integral