Answer
$\int_{-\pi/4}^{\pi/4}\frac{t^4~tan~t}{2+cos~t}~dt = 0$
Work Step by Step
The function $~~tan~t~~$ is an odd function while $~~t^4~~$ is an even function. Then $~~t^4~tan~t~~$ is an odd function.
The function $~~2+cos~t~~$ is an even function. Then $~~\frac{t^4~tan~t}{2+cos~t}~~$ is an odd function.
Therefore:
$\int_{-\pi/4}^{\pi/4}\frac{t^4~tan~t}{2+cos~t}~dt = 0$
