The value of $\frac{\int_0^{\pi / 2}(\sin x)^{\sqrt{2}+1} d x}{\int_0^{\pi / 2}(\sin x)^{\sqrt{2}-1} d x}$ is $........$

If $\int \frac{1}{\cos 4x \cos 2x} dx = \frac{1}{2\sqrt{2}} \log \left(\frac{1+f(x)}{1-f(x)}\right) - \frac{1}{2} \log g(x) + C$,then find the value of $g\left(\frac{\pi}{6}\right) - \sqrt{2} f\left(\frac{\pi}{6}\right)$.

If $\int \frac{a \cos x+3 \sin x}{5 \cos x+2 \sin x} d x=\frac{26}{29} x-\frac{k}{29} \log |5 \cos x+2 \sin x|+c$,then $|a+k|=$

If $\int \frac{3 \cos x-2 \sin x}{4 \sin x+5 \cos x} d x=A \log |5 \cos x+4 \sin x|+B x+c$,then $A$ and $B$ are

Integrate the function: $\frac{1}{\sqrt{\sin ^{3} x \sin (x+\alpha)}}$

Let a function $h(x)$ be defined as $h(x) = 0$, for all $x \ne 0$. Also $\int_{-\infty}^{\infty} h(x) \cdot f(x) \, dx = f(0)$, for every function $f(x)$. Then the value of the definite integral $\int_{-\infty}^{\infty} h'(x) \cdot \sin x \, dx$ is