On a classic integral

Evaluate the integral:

$$I=\int_{-\pi/2}^{\pi/2}\frac{1}{2007^x +1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x +\cos^{2008}x}\, {\rm d}x$$

Solution



 We are applying the classic sub. $u=-x$ hence if $I$ denotes our integral then:

$$\begin{aligned}
I &=\int_{-\pi/2}^{\pi/2}\frac{1}{2007^x +1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x +\cos^{2008}x}\, {\rm d}x \\
 &\overset{u=-x}{=\! =\! =\! =\!} \;\; \int_{-\pi/2}^{\pi/2}\frac{2007^x}{1+2007^x} \cdot \frac{\sin^{2008}x}{\sin^{2008}x +\cos^{2008}x}\, {\rm d}x  \\
 &= I
\end{aligned}$$

Adding the two integrals together we get that:

$$2I= \int_{-\pi/2}^{\pi/2} \frac{\sin^{2008}x}{\sin^{2008}x +\cos^{2008}x}\, {\rm d}x \Rightarrow I= \int_{0}^{\pi/2} \frac{\sin^{2008}x }{\sin^{2008}x + \cos^{2008}x}\, {\rm d}x$$

For the last integral we apply the classic sub $u= \frac{\pi}{2} -x$ and by adding again the two integrals we get that the initial integral is evaluated to $\pi/4$.