1. ## Definite integral

Proof, without calculation, that:

$\int^\pi_{-\pi}(4arctane^x-\pi) dx = 0$

I was thinking to show that the integrand is an odd function and then, since the interval is symmetrical, it means that the definite integral is zero. But how do I show that $4arctane^x-\pi$ is an odd function?

2. ## Re: Definite integral

Use the fact that the derivative of an odd function is even (and that the reverse implication also is true under certain conditions). Derive the integrand and see if you can conclude that the derivative indeed is even.

3. ## Re: Definite integral

What are the "certain conditions"?

4. ## Re: Definite integral

Integrate $f'(x) = f'(-x)$ and see what you come up with.