Can u substitution be used for Integral 3x(cos(2x))^2dx?

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Homework Help Overview

The discussion revolves around the use of substitution in integrals, specifically focusing on the integral of the function 3x(cos(2x))^2dx. Participants explore various approaches to tackle the problem, including the application of trigonometric identities and substitution methods.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the potential use of double angle identities and substitution techniques. There is an exploration of rewriting the integral and considering different substitutions, such as u=sin(x) and u=cos(x). Questions arise about the correctness of the transformations and the next steps in the integration process.

Discussion Status

Some participants have provided guidance on rewriting the integral and suggested different substitution strategies. There is an ongoing exploration of various interpretations and methods, with no explicit consensus reached on the best approach for the integral involving 3x(cos(2x))^2.

Contextual Notes

Participants are navigating through the complexities of integration techniques, including the implications of using different substitutions and the need for clarity on the transformations involved. There is a mention of confusion regarding the application of these methods.

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Homework Statement



integral 31(cos^2x)(sin(2x)dx

Homework Equations





The Attempt at a Solution


I am so lost on this problem... Any suggestions would be great
 
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Try looking up the double angle identities and use them to re-write sin(2x). Then try integration by substitution.
 
So if the double angle for sin(2x) is 2sinxcosx... the problem would be rewritten as integral 31cos^2x(2sinxcosx)dx? Then what? Sorry I am still confused

Thanks for the help!
 
Now collect the cos(x) terms together. Then what do you have?
 
So it would become integral 31(2sinxcos^3)... Then make u=sinx du=cosxdx... Which would lead to 31 integral 2sinxcos^2xcosxdx... Then 31 integral 2sinx(1-sin^2x)du... 31 integral 2u(1-u^2)du... Eventually leading to 31u^2-31/2u^4... And then 31sin^2x-31/2sin^4x.. How does that look??
 
It's much simpler if you try u=cos(x), but I think your solution is correct.
 
Last edited:
Awesome! Thank you so much. I have another question if you don't mind.. . For Integral 3x(cos(2x))^2dx could I use u substitution with the 3x
 

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