Integrating cos(n arcsin x) * xm dx

[Zoe-b]

Homework Statement

Part of a question has brought up this problem:

find the integral of cos(n arcsin x) * x^m dx
where n and m are positive integers > 1.

Homework Equations

Not sure- if the n wasn't there I would simplify it to (1-x²)^1/2 * x^m which would make it easier but as it is I'm not sure how to proceed.
Using a maths programme I have it seems to be able to integrate it only if n is even.. but I have no idea why this is or how to get to the answer.

The Attempt at a Solution

Haven't got very far at all as you can see, sorry.
I already did a substitution from cos(u) * cos (nu) * (sin u)^m du to the above where x = sin u.

 

[mighty2000]

But I'm not sure where to go from there.

Hello, thank you for posting your question on this forum. Let me try to help you find a solution for this integral.

First, let's start by simplifying the expression inside the integral. We can use the trigonometric identity cos(narcsin x) = (1-x^2)^(n/2) to rewrite the expression as:

(1-x^2)^(n/2) * x^m dx

Now, let's make a substitution u = 1-x^2, which means du = -2x dx. We can rewrite the integral as:

-1/2 * ∫ u^(n/2) * u^(m/2 - 1) du

Using the power rule for integration, we get:

-1/2 * (u^(n/2 + m/2) / (n/2 + m/2)) + C

Substituting back u = 1-x^2 and simplifying, we get:

-1/2 * ((1-x^2)^(n/2 + m/2) / (n/2 + m/2)) + C

This is the general solution for the integral. Now, let's consider the case where n is even. In this case, we can rewrite the expression as:

cos(narcsin x) = (1-x^2)^(n/2) = ((1-x^2)^(1/2))^n = sin(arcsin x)^n = x^n

Using this, we can simplify the integral as:

∫ x^(n+m) dx = x^(n+m+1) / (n+m+1) + C

I hope this helps you understand the problem and find a solution. Let me know if you have any further questions. Keep up the good work!