Reduction formula and Walli's Formula

If $I\left(m,n\right)=\underset{0}{\overset{1}{\int }}{t}^m{\left(1+t\right)}^{n}dt$, then the expression for $I\left(m,n\right)$ in terms of $I\left(m+1,n-1\right)$ where $m, n\in N$ is

If $I_n= \underset{0}{\overset{\frac{\pi }{4}}{\int }}{\tan }^n\theta d\theta$, then the value of $I_n+I_{n-2}$ is equal to

Let $\mathbb{N}$ be the set of natural numbers. For $n\in \mathrm{\mathbb{N}},$ define $I_n={\int }_0^{\pi }\frac{x{\sin }^{2n}\left(x\right)}{{\sin }^{2n}\left(x\right)+{\cos }^{2n}\left(x\right)}dx.$

Then for $m,n\in \mathrm{\mathbb{N}}$

Let $f\left(x\right)=\left\{\begin{array}{cc}\frac{x}{\sin x},& x\in (0,1)\\ 1,& x=0\end{array},\right.$ consider the integral $I_n=\sqrt{n}{\int }_0^{1/n}f\left(x\right)e^{-nx}dx$. Then $\underset{n\to \infty }{\lim }{I}_n$

If $I_n={\int }_1^e{\left(\ln x\right)}^{n}dx, n\in I^{+}$, then $I_4$ equals

Find the value of ${\int }_0^{\pi }\frac{\sin 37x}{\sin x}dx$?

Find the value of ${\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}{\sin }^2{\mathrm{xcos}}^3\mathrm{xdx}$ 

If $I_n=\underset{0}{\overset{\pi /4}{\int }}{\tan }^n\left(x\right)dx$ then $I_2+I_4,I_3+I_5,I_4+I_6,...$ are in ?

If $I_n=\int {\tan }^{n}xdx$ for $n\ge 2$, then $I_n+I_{n-2}$ is equal to

If $I_n={\int }_{-\pi }^{\pi }\frac{\sin nx}{\sin \mathit{x}}dx,n\in N$, then which of the following options are correct ?

${\int }_0^{\infty }{\text{x}}^{2\text{n}+1}·{\text{e}}^{-{\text{x}}^2}\text{dx}$ is equal to $\left(\text{n }ϵ\text{ N}\right)$