$\int_{-\pi/4}^{\pi/4} \,(\sqrt{2})^{1/3}\, (cos\,\theta\,sin\,\frac{\pi}{4})+ sin\,\theta\,cos\,\frac{\pi}{4})^{1/3}d\theta \hspace{5cm}$
$\int_{-\pi/4}^{\pi/4} \,(\sqrt{2})^{1/3}\, (sin\,(\theta+\frac{\pi}{4}))^{1/3}d\theta \hspace{13cm} \\ $

Put,

$\theta+\frac{\pi}{4}=t \hspace{0.3cm}=\gt\hspace{0.3cm} d\theta=dt \hspace{13cm} $

when,

$\theta=-\frac{\pi}{4},t=0,\theta=\frac{\pi}{4},t=0 \hspace{0cm} $

$\sqrt{2}^{1/3} \int_{0}^{\pi/2} sin^{1/3}t \,dt=\frac{(\sqrt{2})^{1/3}}{2}\beta (\frac{1/3+1}{2},\frac{1}{2})$

$\frac{\sqrt{2}^{1/3} }{2} \beta (\frac{4}{3*2},\frac{1}{2})=\gt\hspace{0.3cm} \frac{\sqrt{2}^{1/3} }{2} \beta (\frac{4}{6},\frac{1}{2})$

$2^{-5/6}\beta(\frac{2}{3},\frac{1}{2})= 2^{-5/6} \frac{\sqrt{2/3}\sqrt{1/2}}{\sqrt{2/3+1/2}}=2^{-5/6} \frac{\sqrt{2/3}}{\sqrt{7/6}} \pi$