Problem Statement: What is the value of $\int_{-\pi/4}^{\pi/4}(\cos t +\sqrt{1+t^{2}}\sin^{3}t\cos^{3}t)dt?$
I am working on an old GRE Math Subject test, and I am having trouble determining which substitution to do for the given integral.
With algebraic substitution, I have $$u = \cos t,\ \ du = -\sin t dt$$ with the resulting integral $$\sin t - \int \sqrt{1+\arccos^{2}u}\ u^{3}(1-u^{2})du.$$ This seems not to simplify things because of the $\sqrt{1+\arccos^{2}u}$ factor.
With trigonometric substitution, we let $$t = \tan\theta,\ \ dt=\sec^{2}\theta d\theta$$ with the resulting integeral $$\sin t + \int \sec^{3}\theta\ \sin^{3}(\tan\theta)\cos^{3}(\tan\theta)d\theta.$$ Then if I try to do an algebraic-substitution from here I believe I will result in the original integral...
Should I be using integration-by-parts for this integral? I did not try this method because I figured it would be far too complicated, if even possible...
Any suggestions for how I should tackle this integral? The correct resulting value is $\sqrt{2}$.
Thank you
The second part is an odd function and integrates to 0 on $[-\pi/4, \pi/4]$.