# Definite integral

• June 12th 2012, 12:21 PM
loui1410
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?
• June 12th 2012, 12:39 PM
thesmurfmaster
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.
• June 12th 2012, 12:50 PM
loui1410
Re: Definite integral
What are the "certain conditions"?
• June 12th 2012, 01:23 PM
thesmurfmaster
Re: Definite integral
Integrate $f'(x) = f'(-x)$ and see what you come up with.