Reduction Formulae for Indefinite Integration

If $I_n=\int {\tan }^{n}xdx$, and $I_0+I_1+2I_2+2I_3+2I_4+I_5+I_6=\sum _{K=1}^n\frac{{\tan }^{K}x}{K}$, then $n=$

Assertion (A): if $I_n=\int {\cot }^{n}xdx$, then $I_6+I_4=\frac{-{\cot }^{5}x}{5}$

Reason (R): $\int {\cot }^{n}xdx=\frac{-{\cot }^{n-1}x}{n}-\int {\cot }^{n-2}xdx$

Find the integral of $\int x^2\cos 2xdx$ using reduction formula.

Find the integral of $\int x^2\cos xdx$ using reduction formula.

Find the integral of $\int x^2\sin 4xdx$ using reduction formula.

Find the integral of $\int x^2\sin 2xdx$ using reduction formula.

Find the integral of $\int x^2\sin xdx$ using reduction formula.

Find the reduction formula for $\int \frac{\cos x}{x^n}dx$.

Find the reduction formula for $\int \frac{\cos ax}{x^n}dx$.

Find the reduction formula for $\int \frac{\sin 3x}{x^n}dx$.

Find the reduction formula for $\int \frac{\sin x}{x^n}dx$.

Find the reduction formula for $\int \frac{\sin ax}{x^n}dx$.

Find the integral of $\int x^3{e}^{x}dx$ using reduction formula.

Find the integral of $\int x{e}^{2x}dx$ using reduction formula.

Find the integral of $\int x^2{e}^{3x}dx$ using reduction formula.

Find the integral of $\int x^2{e}^{2x}dx$ using reduction formula.

Find the integral of $\int x^2{e}^{x}dx$ using reduction formula.

The value of $\int \left(\sin 101x\right).{\sin }^{99}xdx$ is $\frac{\sin \left(100x\right) {\sin }^{100}x}{k+5}$, then $\frac{k}{19}$ is

If ${\int }_0^1\ln x·\ln \left(1-x\right)dx=p-\frac{{\pi }^q}{r}, p,q,r\in N,$ then $p+q+r$ is equal to $\left[\mathrm{use}\underset{\mathrm{n}\to \infty }{\lim } \sum _{\mathrm{r}=1}^{\mathrm{n}}\frac{1}{r^2}=\frac{{\mathrm{\pi }}^2}{6}\right]$

If $I_n=\int {\left(\log x\right)}^{n}dx$, then $I_n+n I_{n-1}$ $=$?