When dealing with complex numbers, does the i term get integrated as well? An example is an integration for e^4i

Question

I am doing complex analysis. I am stuck on integrating sin^2(4x), from 0 to 2pi. I have to show the integration in exponential form.

Answer 1

In general, the answer to your first question is Yes.  The imaginary term is integrated along with the real term, and the same rules apply.  i is treated like any other constant.

 

Integrating sin² 4x is done as follows:

 

Remember that cos 2x = 1 - 2 sin² x.  So sin² x = (1 - cos 2x)/2

 

So sin² 4x = (1 - cos 8x) / 2

 

So ∫ sin ² 4x dx = ∫ (1 - cos 8x) / 2 dx =1/2 ( ∫ 1 - cos 8x) dx = 1/2 (x - 1/8 sin 8x) + C

 

Then evaluate the definite integral in the usual way.

 

Answer 2

When dealing with integrals that have complex numbers we treat the imaginary number as a constant.

 

See

 

http://www.mathnotes.org/?pid=109#?pid=109