# Difficult Integral

1. Jan 24, 2013

### iRaid

1. The problem statement, all variables and given/known data
$$\int (arcsin x)^{2}$$

2. Relevant equations

3. The attempt at a solution
$$u=arcsin x$$ $$du=1/\sqrt{1-x^{2}}dx$$
$$v=?$$ $$dv=arcsin x dx$$

2. Jan 24, 2013

### Dick

Re: Integral

To integrate arcsin(x)dx do parts again. u=arcsin(x) dv=dx.

3. Jan 24, 2013

### iRaid

Re: Integral

Ok then I get:
$$xarcsinx-\int x/(\sqrt{1-x^{2}})$$

4. Jan 24, 2013

### Dick

Re: Integral

The second term can be integrated with a simple substitution.

5. Jan 24, 2013

### iRaid

Re: Integral

Ok after all that:
$$\int arcsinx=xarcsinx+\sqrt{1-x^{2}}$$
Then putting in for v, I end up with:
$$(arcsinx(arcsinx+\sqrt{1-x^{2}})-\int \frac{(xarcsinx+\sqrt{1-x^{2}})}{(\sqrt{1-x^{2}})}$$

Do I do another u-substition for the integral, u=arcsinx? But I don't know what to do with the x infront of it.

6. Jan 24, 2013

### Zondrina

Re: Integral

No no!

Let u = 1 - x2 so that du = -2x dx which means -du/2 = x dx.

What does that second integral x / sqrt(1-x2) become now?

7. Jan 24, 2013

### iRaid

Re: Integral

That doesn't work or I'm not understanding

8. Jan 24, 2013

### Zondrina

Re: Integral

$\int \frac{x}{\sqrt{1-x^2}} dx$

$u = 1-x^2 → - \frac{1}{2} du = xdx$

$\int \frac{x}{\sqrt{1-x^2}} dx = - \frac{1}{2} \int \frac{1}{\sqrt{u}} du$

9. Jan 24, 2013

### Dick

Re: Integral

Split the integral into two parts. Looks like you need another round of integration by parts on the first piece.

10. Jan 24, 2013

### iRaid

Re: Integral

The integral I'm trying to solve is:
$$\int \frac{xarcsinx+\sqrt{1-x^2}}{\sqrt{1-x^{2}}}dx$$

11. Jan 24, 2013

### Zondrina

Re: Integral

Oh my apologies, I thought :

Was what you were having trouble with.

12. Jan 24, 2013

### iRaid

Re: Integral

Lol all good, I was like am I being stupid or what?

13. Jan 24, 2013

### iRaid

Re: Integral

This just never ends omg.

14. Jan 24, 2013

### Dick

Re: Integral

You are almost there. It's downhill from here.

15. Jan 24, 2013

### iRaid

Re: Integral

I don't know the derivative of xarcsinx

16. Jan 24, 2013

### Dick

Re: Integral

u=arcsin(x) dv=xdx/sqrt(1-x^2). Integrating that should look familiar.

17. Jan 24, 2013

### iRaid

Re: Integral

Yeah I got it,
$$arcsinx(arcsinx+\sqrt{1-x^{2}})-(\frac{x(arcsinx)^{2}}{2}+x)$$

18. Jan 24, 2013

### Dick

Re: Integral

Not right I'm afraid. I think you might have goofed up the last integration by parts. Check it.

19. Jan 24, 2013

### iRaid

Re: Integral

This is ridiculous so much work just to get the wrong answer lol.

I got what I did wrong, it was the last integration by parts.

20. Jan 24, 2013

### Dick

Re: Integral

Now that you know the steps you just have to make sure everything is right. You're close.