Integration by Parts

What would be the value of  $\int \left(x\log 2x\right)dx$

Integrate  ${\displaystyle \int \frac{x^3+\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}3x+2}{{(x^2+1)}^2(x+1)}dx}$

Evaluate: $\int \log xdx$

$\int x{e}^x\sin xdx=\_\_\_\_\_$

$\int \frac{1}{x^3}{e}^{\frac{-1}{x}}dx=$

$\int \left(\frac{1}{{\log }_e\left(x\right)}-\frac{1}{{\left({\log }_e\left(x\right)\right)}^2}\right)dx=$

If $f\left(x\right)=\int \frac{e^{\cos x}\left(2{\cos }^{3}x\sin x+\sin x{\cos }^{2}x-4\sin x\cos x\right)dx}{2{\left(1-\cos x\right)}^{{\displaystyle \frac{3}{2}}}}$ and given that $f\left(\frac{\pi }{2}\right)=0$ then which of the following is true?

The value of $\int {\tan }^{-1}xdx$, is equal to

If $\frac{\mathrm{d}}{\mathrm{d}\mathrm{x}}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\cos \mathrm{x}+\sin \mathrm{x}$ and $\mathrm{f}\left(0\right)=2$ , then $\mathrm{f}\left(\mathrm{x}\right)$ is equal to

Integrate the following function w.r. t. $x$:

$\frac{e^x}{x}\left[x·(\log x{)}^2+2·\log x\right]$

Evaluate $\int x{e}^{x}dx$.

Find: $\int e^x\sin x dx$

Find $\int e^{ax}·\sin (bx+c)dx$

Find $\int \frac{(x-3)e^x}{(x-1{)}^3}dx$

Find $\int x {\sec }^{2}x dx$.

Evaluate $\int x\log xdx$

Let $\ln x$ denote the logarithm of $x$ with respect to the base $e$ Let $S\subset R$ be the set of all points where the function $\ln \left(x^2-1\right)$ is well-defined. Then, the number of functions $f:S\to R$ that are differentiable, satisfy $f^{\text{'}}\left(x\right)=\ln \left(x^2-1\right)$ for all $x\in S$ and $f\left(2\right)=0,$ is