Throwing everything I can at this one

1. Sep 23, 2011

1MileCrash

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

First I attempted by parts, that got nowhere. Then, I attempted a u substitution for u = arcsin x, and when that did nothing I attempted one for u = arcsinx^2.

I have no idea what to do with this one. How should I approach the problem?

Last edited: Sep 23, 2011
2. Sep 23, 2011

gb7nash

Is the problem $\int \frac{(arctanx)^{2}}{\sqrt{1-x^{2}}} dx$ or $\int \frac{(arcsinx)^{2}}{\sqrt{1-x^{2}}} dx$?

3. Sep 23, 2011

1MileCrash

4. Sep 23, 2011

gb7nash

Your original intuition looks fine. Set u = arcsinx. du = ...?

5. Sep 23, 2011

1MileCrash

My original intuition of integration by parts?? I didn't think that was the preferred method because I'd have to integrate (1-x^2)^(-1/2) which would just complicate it

6. Sep 23, 2011

1MileCrash

Ok wow, this is an extremely simple substitution problem, provided you actually remember the derivative has a square root in the denominator.

Try doing it thinking it's like the arctan derivative and you will see my frustration.

7. Sep 23, 2011

gb7nash

If it was, I have no idea how to solve that. You probably won't make that mistake again.

8. Sep 23, 2011

Ignea_unda

Numerically.

9. Sep 23, 2011

Staff: Mentor

Not really a solution for an indefinite integral...

10. Sep 23, 2011

Ignea_unda

Touche, I guess...didn't look that closely and missed the lack of limits.

11. Sep 23, 2011

1MileCrash

Integration is getting really difficult for me with all the new techniques I had to learn in like 2 days.

How do you guys decide which method to try first?

12. Sep 23, 2011

rock.freak667

Usually depending on how difficult it looks, you should try to do substitution before you attempt integration by parts.

For example, in your question, you can see a relation between arcsin(x) and 1/sqrt(1-x^2), substitution might work here.

13. Sep 23, 2011

1MileCrash

I have another question.

One of my favorite integrals to solve is actually integral of the arctangent. I like it because it's kind of neat, integrate by parts once, then u-substitution once, it looks good on paper.

However, you need to know it's derivative when you call it u for integration by parts.

I just have it memorized. I don't really intuitively understand how one arrives at the derivative of arctangent. Is it just something that is observable or can it actually be shown?