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.
We can use Euler's identity instead: At this point, it would be possible to change back to real numbers using the formula e2ix + e−2ix = 2 cos 2x.
Either method gives 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 cos x is the real part of eix, we know that The integral on the right is easy to evaluate: Thus: In general, this technique may be used to evaluate any fractions involving trigonometric functions.
, the result is the integral of a rational function: One may proceed using partial fraction decomposition.