If $2 f(x)-3 f\left(\frac{1}{x}\right)=x$,then $\int_1^e f(x) d x=$

Prove that $\int_{0}^{1} \sin^{-1} x \, dx = \frac{\pi}{2} - 1$.

Let $[.]$ denote the greatest integer function. If $\int_0^{e^3}\left[\frac{1}{e^{x-1}}\right] d x=\alpha-\log _e 2$,then $\alpha^3$ is equal to . . . . . . .

If $\int_{0}^{k} \frac{dx}{2 + 18x^2} = \frac{\pi}{24}$,then the value of $k$ is

$\int_{0}^{\frac{\pi}{2}} \frac{\sin x \cos x}{1+\sin ^{4} x} d x=$

Let $f:[0,1] \rightarrow [0,1]$ be a continuous function such that $x^2+(f(x))^2 \leq 1$ for all $x \in [0,1]$ and $\int_0^1 f(x) dx = \frac{\pi}{4}$. Then,$\int_{\frac{1}{2}}^{\frac{1}{\sqrt{2}}} \frac{f(x)}{1-x^2} dx$ equals