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

  • Thread starter Jimbo57
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    Integration
In summary, the problem is to integrate x^2(x+1)^1/2 from x=0 to x=3, and the solution involves using the substitution u=x+1 and integration by parts. The final integrand is (u-1)^2u^1/2.
  • #1
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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  • #2
Try integration by parts.
 
  • #3
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?
 
  • #4
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!
 

1. What is definite integration?

Definite integration is a mathematical concept used to find the exact area under a curve between two given points. It involves finding the antiderivative of a function and evaluating it at the upper and lower limits of integration.

2. How is definite integration different from indefinite integration?

Definite integration involves finding a specific numerical value, while indefinite integration involves finding a general formula for the antiderivative of a function. In definite integration, the limits of integration are specified, while in indefinite integration, they are not.

3. What is the purpose of definite integration?

The main purpose of definite integration is to calculate the exact area under a curve, which can have various real-world applications in fields such as physics, engineering, and economics. It can also be used to find the volume of a solid with a known cross-sectional area.

4. How is definite integration used in real life?

Definite integration is used in many real-life scenarios, such as calculating the work done by a force, finding the center of mass of an object, and determining the average value of a function. It is also used in economics to determine the total revenue or profit of a business.

5. Are there any limitations to definite integration?

Yes, there are some limitations to definite integration. It can only be applied to continuous functions, and certain functions may not have an analytical antiderivative. Additionally, the accuracy of the result depends on the precision of the integration method used.

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