Kill two birds with a stone

Integration using complex variable

Surprisingly, the use of complex variables can help to solve integration, in a neat and compact way. We like to illustrate this in the following example. Two integrals can be evaluated at the same time. To begin, I suppose you know the Euler formula:

e^ix = cos x + i sin x

Example

Evaluate

C = ò e^2x cos 4x dx, S = ò e^2x sin 4x dx.

Solution

C + iS = ò e^2x (cos 4x + i sin 4x) dx

= ò e^2x e^4ix dx

= ò e^(2+4i)x dx

(for simplicity we don’t write the integrating constant)

Comparing the real and imaginary parts, we get:

You’ve got two birds with a stone!

Note

This method can be used to find the integrals of the form:

C = ò e^ax cos bx dx, S = ò e^ax sin bx dx.