Level 30 Integral

$\begin{aligned} I & = \int_0^\frac \pi 2 \int_0^\frac \pi 4 (\sin x + \cos y)^2\ dx \ dy & \small \color{#3D99F6} \text{Swapping the order of integration} \\ & = \int_0^\frac \pi 4 \int_0^\frac \pi 2 (\sin x + \cos y)^2\ dy \ dx & \small \color{#3D99F6} \text{Using }\int_a^b f(x)\ dx = \int_a^b f(a+b-x)\ dx \\ & = \int_0^\frac \pi 4 \frac 12 \int_0^\frac \pi 2 \left((\sin x + \cos y)^2 + (\sin x + \sin y)^2\right) dy \ dx \\ & = \frac 12 \int_0^\frac \pi 4 \int_0^\frac \pi 2 \left(2\sin^2 x + 2\sin x(\cos y+\sin y) + 1 \right) dy \ dx \\ & = \frac 12 \int_0^\frac \pi 4 \bigg[2y\sin^2 x + 2\sin x(\sin y-\cos y) + y \bigg]_0^\frac \pi 2 \ dx \\ & = \frac 12 \int_0^\frac \pi 4 \left(\pi \sin^2 x + 4\sin x + \frac \pi 2 \right) dx \\ & = \left[\frac \pi 4 \left(x - \frac {\sin 2x}2\right) - 2\cos x + \frac \pi 4 x\right]_0^\frac \pi 4 \\ & = \frac {\pi^2-\pi}8 + 2 - \sqrt 2 \end{aligned}$

Therefore $a=\boxed 2$ .