Undefined Integral ArcSin(x) / x^2 dx

[nikola]

Hello,
The problem:
Undefined Integral ArcSin(x) / x^2 dx

At first I thought this integral is very simple but I got in trouble so all your help is welcomed.
I tried part integration using u=ArcSin(x) and dv=x^(-2) dx and then got

du=1/sqrt(1-x^2) dx and v=-1/x
using part integration
-ArcSin(x)/x + Integral 1/ x sqrt(1-x^2) dx //this second integral is bit of a problem for me.

Thank You
All the best, Nikola, Europe

 

[galactus]

You did good, but you're having trouble with:

\(\displaystyle \int\frac{1}{x\sqrt{1-x^{2}}}dx\)

You can use trig sub.

Let \(\displaystyle x=sin(t), \;\ dx=cos(t)dt\)

When you make the subs it whittles down to:

\(\displaystyle \int{csc(t)}dt\)

Now, can you proceed?.

 

[royhaas]

In the second integral, use \(\displaystyle x = sin(u)\).

 

[nikola]

Solved!
Thank you a lot. Now I now what was the main problem.I have tried trig sub but I have hardly ever heard a function called cosecant(x) == 1/sin(x) so that integral 1 / sin(x) dx was a problem for me again. Csc (x) isn't used very often in Europe says Google too . I am making progress.
Thanks again!

 

[galactus]

Keep in mind that \(\displaystyle \frac{1}{sin(x)}=csc(x), \;\ \frac{1}{cos(x)}=sec(x), \;\ \frac{1}{tan(x)}=cot(x)\)

incase you need them somewhere down the road.