# 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?