1. ## Integration

Hello everyone,

Im having difficulties with this integral.

Integral x csc x cot x dx

It cant be subsitution because dy/dx of csc x is -cscx and cotx.
It got to be integration by parts but i have x csc x and cot x,
anyone can give me a little bit so I can start it off. Thanks.

2. Originally Posted by ff4930
Integral x csc x cot x dx
$\displaystyle \int x\csc x\cot x\,dx=\int\frac{x\cos x}{\sin^2x}\,dx$

Define $\displaystyle \alpha=\sin x$

Next step is integration by parts.

3. ugh, Im still stumped. I know how you got to where you are tho.

Can I move the sin^2x up and become (sin^2x)^-1?
let u be xcosx and dv = (sin^2x)^-1

4. $\displaystyle \int\frac{x\cos x}{\sin^2x}\,dx$

$\displaystyle \alpha=\sin x\implies d\alpha=\cos x\,dx,$ the integral becomes to

$\displaystyle \int\frac{x\cos x}{\sin^2x}\,dx=\int\frac{\arcsin\alpha}{\alpha^2} \,d\alpha$

Now let's apply integration by parts, so set $\displaystyle u=\arcsin\alpha\implies du=\frac1{\sqrt{1-\alpha^2}}\,d\alpha$ and $\displaystyle dv=\alpha^{-2}\,d\alpha\implies v=-\frac1\alpha,$ the new integral becomes to

$\displaystyle \int\frac{\arcsin\alpha}{\alpha^2}\,d\alpha=-\frac{\arcsin\alpha}\alpha+\int\frac1{\alpha\sqrt{ 1-\alpha^2}}\,d\alpha$

The remaining integral is easy to solve, so its answer is

$\displaystyle \ln|\alpha|-\ln\left(\sqrt{1-\alpha^2}+1\right)$

Finally

$\displaystyle \int\frac{\arcsin\alpha}{\alpha^2}\,d\alpha=-\frac{\arcsin\alpha}\alpha+\ln|\alpha|-\ln\left(\sqrt{1-\alpha^2}+1\right)$

Back substitute and you are done.