What is the solution to ∫x^2(x+1)^1/2 for x=0 x=3?

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Homework Help Overview

The problem involves integrating the function x^2(x+1)^(1/2) over the interval from x=0 to x=3. The subject area pertains to calculus, specifically integration techniques.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts substitution with u=x+1 and expresses uncertainty about handling the x^2 term. Some participants suggest integration by parts as a potential method. There is a discussion about the integrand after substitution, with one participant reflecting on the simplification of the expression.

Discussion Status

The discussion is ongoing, with participants exploring different integration techniques. Some guidance has been offered regarding the substitution method, and there is recognition of a previously solved similar problem, indicating a productive exchange of ideas.

Contextual Notes

Participants note the challenge posed by the x^2 term and the realization that they had encountered a similar problem in their textbook, which may influence their approach to the current problem.

Jimbo57
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Homework Statement



Integrate x^2(x+1)^1/2 for x=0 x=3

Homework Equations





The Attempt at a Solution



I start with substitution u=x+1
and du=dx

I have no clue where to go from here. How do I take care of that x^2?

This looks like it would be simple but it's giving me a hell of a time.

Jim
 
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Try integration by parts.
 
Jimbo57 said:

Homework Statement



Integrate x^2(x+1)^1/2 for x=0 x=3

Homework Equations



The Attempt at a Solution



I start with substitution u=x+1
and du=dx

I have no clue where to go from here. How do I take care of that x^2?

This looks like it would be simple but it's giving me a hell of a time.

Jim
That substitution, u=x+1, should work fine, and allow you to complete the integration.

What do you get for the integrand when you use that substitution?
 
Looking through my textbook I realized that we already solved this one as an indefinite integral... kind of embarrassing.

Dealing with the x^2 was the hardest, but the integrand is:

u=x+1
du=dx
x^2=(u-1)^2 <---- Much easier looking at it now.

∫(u-1)^2u^1/2du

Thanks for the help as always!
 

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