Double Integral

Evaluate the following double integral:

$$\mathcal{J}=\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{\left ( \cos x-\sin x \right )^{y-2}}{\left ( \sin x+\cos x \right )^{y+2}}\,dy\,dx$$

Solution:



We are invoking Fubini's theorem for interchanging the integrals. Hence:

$$\begin{aligned}
\mathcal{J}&=\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{\left ( \cos x-\sin x \right )^{y-2}}{\left ( \sin x+\cos x \right )^{y+2}}\,dy\,dx \\
 &= \int_{\pi/2}^{\pi}\int_{0}^{\pi/4}\frac{\left ( \cos x-\sin x \right )^{y-2}}{\left ( \sin x+\cos x \right )^{y+2}}\,dx\,dy\\
 &\overset{u=\pi/4-x}{=\! =\! =\! =\! =\!} \frac{1}{4}\int_{\pi/2}^{\pi}\int_{0}^{\pi/4}\frac{\left ( \tan x \right )^{y-2}}{\left ( \sec x \right )^{y+2}}\, dx \,dy\\
 &= \frac{1}{4}\int_{\pi/2}^{\pi}\int_{0}^{\pi/4}\left ( \tan x \right )^{y-2}\sec^4 x \, dx\,dy\\
 &= \frac{1}{4}\int_{\pi/2}^{\pi}\frac{2y}{y^2-1}\,dy \\
 &=\frac{1}{4}\left [ \ln 4 +\ln \left ( \pi^2-1 \right )+\ln \left ( \pi^4-4 \right ) \right ]
\end{aligned}$$