$\int \cos^{-3/7} x \sin^{-11/7} x \, dx = $

If $\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x=A \sin 2 x+B$,then $A$ is equal to

If $\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B \sqrt{\cos \theta-\cot x \sin \theta}+C,$ where $C$ is the integration constant,then $AB$ is equal to

If $I_1 = \int \sin^6 x \, dx$ and $I_2 = \int \cos^6 x \, dx$, then $I_1 + I_2 = $

Let $I(x) = \int \sqrt{\frac{x+7}{x}} \, dx$ and $I(9) = 12 + 7 \log_e 7$. If $I(1) = \alpha + 7 \log_e(1 + 2\sqrt{2})$,then $\alpha^4$ is equal to $..........$.

If $\int e^u \sin 2x \, dx$ can be expressed in terms of elementary functions of $x$,then $u$ can be: