Calculus/Integration techniques/Integration by Complexifying

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Integration techniques/Integration by Complexifying

This technique requires an understanding and recognition of complex numbers. Specifically Euler's formula:

Recognize, for example, that the real portion:

Given an integral of the general form:

We can complexify it:

With basic rules of exponents:

It can be proven that the "real portion" operator can be moved outside the integral:

The integral easily evaluates:

Multiplying and dividing by  :

Which can be rewritten as:

Applying Euler's forumula:


Taking the Real part of this expression: