Calculus/Integration techniques/Integration by Complexifying

From Wikibooks, open books for an open world
Jump to: navigation, search
← Integration techniques/Integration by Parts Calculus Integration techniques/Partial Fraction Decomposition →
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 (1-2i):

Which can be rewritten as:

Applying Euler's forumula:

Expanding:

Taking the Real part of this expression:

So: