Throwing everything I can at this one

[1MileCrash]

Homework Statement

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

Homework Equations

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?

 

[gb7nash]

Homework Statement

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

Homework Equations

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?

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

 

[1MileCrash]

arcsin, definitely. Sorry about that.

 

[gb7nash]

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

 

[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

 

[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.

 

[gb7nash]

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

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

 

[Ignea_unda]

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

Numerically.

 

[Mark44]

Numerically.

Not really a solution for an indefinite integral...

 

[Ignea_unda]

Not really a solution for an indefinite integral...

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

 

[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?

 

[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.

 

[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?