Solving Integration Problems: Tips and Tricks for Integrating x arccos(x)dx

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Homework Help Overview

The problem involves integrating the function x arccos(x) with respect to x. Participants are exploring various methods to approach this integration challenge.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts integration by parts but encounters difficulties with the resulting integral. Some participants suggest substitutions, such as x=sin(u) and u=arccos(x), to simplify the problem. Others consider the implications of these substitutions on the integral's form.

Discussion Status

The discussion is active, with participants offering different substitution methods and confirming each other's interpretations. There is no explicit consensus on the best approach yet, but multiple lines of reasoning are being explored.

Contextual Notes

Participants are working within the constraints of a homework assignment, which may limit the information they can share or the methods they can use. There is some confusion regarding the integral's setup, particularly the expression inside the integral.

colorado
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This is a problem from my homework set.

I'm so close but I'm tangled up at the end...

Integrate/ x arccos(x)dx

So far I am at

(x^2)/2 arccosx - Int/ (x^2)/ (2 sqrt(x^2 - 1))

Can figure out how to integrate this part.
 
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I hope you mean sqrt(1-x^2). I would try a substitution like x=sin(u).
 
You are correct. Thankyou!
 
I'm not sure I'm reading this right. Is the part inside the integral,

x * arccos(x)?

if so, what about the substitution u = arccos(x), so that x=cos(u), and dx = -sin(u) du? Then it becomes the integral of -u*sin(u)*cos(u) du

I'd probably try repeated integration by parts to solve the new integral.
 
Never mind, seems it was answered while I was typing :)
 

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