Solve Integration: x^5cos(x^3) dx

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Discussion Overview

The discussion revolves around the integration of the function \( \int x^5 \cos(x^3) \, dx \). Participants explore various methods for solving this integral, including substitution and integration by parts.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant requests help with the integration, indicating a lack of progress.
  • Another participant suggests that the original poster should demonstrate their efforts and consider using substitution to simplify the integrand.
  • A participant shares their attempts at integration by parts, noting difficulties with both choices of \( u \) and \( dv \) and expressing uncertainty about how to proceed.
  • Another suggestion is made to use the substitution \( u = x^3 \) and to apply integration by parts in the new variable.
  • There is a reiteration of the suggestion to look for a substitution to simplify the integrand before applying integration by parts.

Areas of Agreement / Disagreement

Participants generally agree on the need for substitution and integration by parts, but there is no consensus on the best approach or substitution to use for this integral. The discussion remains unresolved regarding the most effective method.

Contextual Notes

Participants express uncertainty about the effectiveness of their proposed substitutions and methods, indicating that the integration process may depend on the choice of substitution and the subsequent steps taken.

mckallin
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Please Help! Integration!

can anyone help me solve the following integration? thanks a lot.

[tex]\int[/tex] [tex]x^5[/tex]cos[tex](x^3)[/tex] dx
 
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you really should show some more effort. ie. show us what you have tried, what you know and tell us what you have problems with.hint: try look for a substitution to make the integrand look nicer, then you can try by parts.
 
I have tried to do it by part.

If I make [tex]u=cos(x^3), dv=x^5 dx[/tex], the grade of x, which is in the [tex]cos(x^3)[/tex], won't be reduce.

If I make [tex]u=x^5, dv=cos(x^3) dx[/tex], I can't solve the [tex]\int cos(x^3) dx[/tex].

I have thought if there is some way to make it look nicer (like [tex]u=x^3[/tex] ),but I still can't work out a better substitution.

Could you give me some more advice?
 
try
[tex]u=x^3,dv=x^2\cos(x^3)dx[/tex]

have you learned partial substitution ?
 
mjsd said:
hint: try look for a substitution to make the integrand look nicer, then you can try by parts.

here I suggested a two-step process,
substitiion: u = x^3 seems ok
then by parts in new variable

then put answer back in x.
 

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