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Calculus/Integration techniques/Integration by Complexifying
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Integration techniques/Integration by Parts Integration techniques/Partial Fraction Decomposition
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 formula:
Expanding:
Taking the Real part of this expression:
So: