Evaluate $\int_{-\pi/4}^{\pi/4}(\cos(t)+\sqrt{1+t^2}\cos(t)^3\sin (t)^3)dt$

Question

Evaluate: $$\int_{-\pi/4}^{\pi/4}(\cos(t)+\sqrt{1+t^2}\cos(t)^3\sin(t)^3)dt$$

I have tried substituting $t=\tan(\theta)$ and breaking up the $\cos(t)^3$ using $\cos(t)^2=1-\sin(t)^2$, but I can't figure it out.

This question is #21 from the Math subject GRE form GR1268.

Answer 1

Think about the parity of the second summand. You're integrating over a symmetric interval around the origin.