Question

How do you take the integral of dy/(y^2 + 4)?

Answer 1

use the good old trig substitutions. When I first encountered them, they made no sense to me, but after putting the book away for a few weeks, I took another look, and it was obvious how useful they were.

Let y = 2 tan u
dy = 2 sec²u
Then
y² = 4 tan²u
y² + 4 = 4tan²u + 4 = 4sec²u

Now we have dy/4sec²u = 2sec²u/4sec²u = 1/2

So, integrating, we get u/2

But, since y = 2tan u, u = tan^-1(y/2)

so, we end up with 1/2 tan^-1(y/2) + C