$\int_0^{\pi / 4} \frac{d x}{\cos ^3(x) \cdot \sqrt{2 \sin (2 x)}}=$

Let $\alpha > 0$. If $\int \limits _0^\alpha \frac{ x }{\sqrt{ x +\alpha}-\sqrt{ x }} dx =\frac{16+20 \sqrt{2}}{15}$,then $\alpha$ is equal to :

Let $I_{1}=\int_{0}^{n}[x] d x$ and $I_{2}=\int_{0}^{n}\{x\} d x,$ where $[x]$ and $\{x\}$ are the integral and fractional parts of $x$ respectively,and $n \in N-\{1\}.$ Then,$I_{1} / I_{2}$ is equal to

$\int_0^{\pi / 2} \sin ^5\left(\frac{x}{2}\right) \cdot \sin x \, dx =$

If $\int_0^{\frac{1}{2}} \frac{x^2}{\left(1-x^2\right)^{\frac{3}{2}}} \,d x=\frac{k}{6}$, then the value of $k$ is

If $f$ is integrable on $[0, a]$,then the function $h$ defined on $[0, a]$ as $h(x) = \int_0^x f(t) dt$ is integrable on $[0, a]$. Which of the following functions is also integrable on $[0, a]$?