Methods of Integration

What would be the value of $\int \frac{x^2+1}{{\left(x-1\right)}^2\left(x+3\right)}dx$

If  ${\displaystyle \int \frac{4e^x+6e^{-x}}{9e^x-4e^{-x}}dx}=Ax+B\log (9e^{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}2x}-4)+C,$  then

$A=\mathrm{...},B=\mathrm{...},\mathrm{and}C=\mathrm{...}$

Evaluate the integral $\int \left(x\sin x\right)dx$.

Find $\int e^{-x}\cos \left(x\right)dx$ using complex numbers.

Consider the following statements

$$\begin{gathered} \left(i\right) \int \log 10 dx=x+c \\ \left(ii\right) \int {10}^{x}dx={10}^x+c \end{gathered}$$

Which of the above statements is/are correct?

If $\int \frac{dx}{x^3{\left(1+x^6\right)}^{{\displaystyle \frac{2}{3}}}}=f\left(x\right){\left(1+x^{-6}\right)}^{\frac{1}{3}}+C$ where $C$ is a constant of integration, then $f\left(x\right)$ is equal to

If $\int \left[\frac{1-x^9}{x\left(1+x^9\right)}\right]dx=A\log \left|x\right|+B\log \left|1+x^9\right|+C$, then the ratio $A:B$ is equal to

Let $I=\int \frac{e^x}{e^{4x}+e^{2x}+1}dx$ and $J=\int \frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx,$ then $J-I$ equals

If $\mathrm{p}, \mathrm{q}$ and $r$ are prime numbers such that ${\mathrm{p}}^2-{\mathrm{q}}^2=\mathrm{r},$ then $\int \frac{\mathrm{dx}}{{\mathrm{x}}^2+\mathrm{px}+\mathrm{q}}$ is

The value of $\int \frac{{\left({\sin }^{-1}x\right)}^4}{\sqrt{1-x^2}}dx$ is:

If ${\mathit{\text{l}}}^{\text{r}}\left(\text{x}\right)$ means $\mathit{\text{log log log}}\text{..... x}$, the log being repeated $\text{r}$ times, then $\int {\left[\mathit{\text{x}}\mathit{\text{l}}\left(\mathit{\text{x}}\right){\mathit{\text{l}}}^2\left(\mathit{\text{x}}\right){\mathit{\text{l}}}^3\left(\mathit{\text{x}}\right)\text{......}{\mathit{\text{l}}}^{\text{r}}\left(\mathit{\text{x}}\right)\right]}^{-1}\mathit{\text{dx}}$ =

$\int \frac{\sqrt{\cot x}}{\sin x\cos x}dx=\dots ..+c;x\ne \frac{n\pi }{2}, n\in Z,\cot x>0.$

The value of the integral $\int \frac{x}{1+x\tan x}dx$ is equal to (Where $C$ is constant of integration)

The partial fractions of $\frac{3x^3-8x^2+10}{{\left(x-1\right)}^4}$ is