Integration by parts with unity factor

Question

Famously, log and arctan can be integrated by parts when you write $\log(x)=\log(x)\cdot1$. Are there any other somewhat elementary examples of that phenomenon?

Answer 1

Yes, the $\arcsin$ and $\arccos$ functions would work as well. E.g.

$$\int \arccos x \; dx = x \arccos x + \int \frac{x}{\sqrt{1-x^2}} dx$$

$$=x \arccos x - \int d(\sqrt{1-x^2})$$

$$=x \arccos x - \sqrt{1-x^2}$$

The inverse hyperbolic functions are other examples.

Answer 2

some other examples are:

$\int(log x)^2dx$

$\int(arcsin)dx$

$\int arcsin(\sqrt x)dx$

$\int (arcsinx)^2dx$

$\int(arccos)dx$

$\int(arctan)dx$

$\int(arcsex)dx$

$\int (arcsecx)^3 dx$

$\int \frac{x-sinx}{1-cosx}dx$

$\int log(1+x^2)dx$

$\int \frac{arcsin(x)}{(1-x^2)^{3/2}}dx$

$\int x^3 e^xdx$

$\int x^3log(2x)dx$

$\int \frac{arcsin(\sqrt x)-arccos(\sqrt x)}{arcsin(\sqrt x)+arccos(\sqrt x)}dx$

$\int \frac{\sqrt {x^2+1}[log(1+x^2)-2logx]}{x^4}dx$

$\int \frac{x^2}{(xsinx + cosx)^2}dx$

Now, have fun doing integrals! :)