# Integrating cos(n arcsin x) * xm dx

• Zoe-b
In summary, the problem is to find the integral of cos(narcsin x) * xm dx, where n and m are positive integers > 1. By simplifying the expression using trigonometric identities and making a substitution, we can rewrite the integral as -(1-x^2)^(n/2 + m/2) / (n/2 + m/2) + C. If n is even, the expression can be further simplified to x^(n+m+1) / (n+m+1) + C.
Zoe-b

## Homework Statement

Part of a question has brought up this problem:

find the integral of cos(n arcsin x) * xm 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 * xm 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.

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!

## 1. What is the formula for integrating cos(n arcsin x) * xm dx?

The formula for integrating cos(n arcsin x) * xm dx is
∫ cos(n arcsin x) * xm dx = (-1)n * (x2 - 1)n/2 * (xm+1) / (m+1) * (1 - x2)1/2 + C,
where n is a constant and m is a positive integer.

## 2. What is the main concept behind integrating cos(n arcsin x) * xm dx?

The main concept behind integrating cos(n arcsin x) * xm dx is using the trigonometric identity cos(n arcsin x) = (x2 - 1)n/2 * (1 - x2)1/2 to simplify the integral into a form that can be easily integrated using standard techniques.

## 3. Is it possible to solve the integral of cos(n arcsin x) * xm dx without using trigonometric identities?

No, it is not possible to solve the integral of cos(n arcsin x) * xm dx without using trigonometric identities. The presence of the trigonometric function cos(n arcsin x) is crucial in simplifying the integral.

## 4. Can the formula for integrating cos(n arcsin x) * xm dx be used for any value of n and m?

Yes, the formula for integrating cos(n arcsin x) * xm dx can be used for any value of n and m, as long as n is a constant and m is a positive integer.

## 5. How can integrating cos(n arcsin x) * xm dx be applied in real-life situations?

Integrating cos(n arcsin x) * xm dx can be applied in real-life situations, such as in solving problems related to motion and oscillations, as well as in modeling physical phenomena involving trigonometric functions. It is also commonly used in the field of engineering for signal analysis and processing.

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