The value of the integral $\int_{0}^{\pi / 2} (\sin^{100} x - \cos^{100} x) dx$ is

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (x^{13} + x \cos x + \tan^{15} x + 1) \, dx$ is . . . . . .

If $I=\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} d x$,then the value of $\int_0^{\frac{\pi}{2}} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x$ is:

The value of $\int\limits_{\frac{1}{2}}^2 \frac{1}{x} \sin \left( x - \frac{1}{x} \right) dx$ is equal to

$\int_0^{\pi /2} {x\cot x\,dx} $ equals

If $\int_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x \, dx}{(1+e^{\sin x})(1+\sin ^4 x)} = \alpha \pi + \beta \log _e(3+2 \sqrt{2})$,where $\alpha, \beta$ are integers,then $\alpha^2+\beta^2$ equals.....................