- #1

Lo.Lee.Ta.

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u = x^3 + 1

du = 3x^2dx

(1/3)du = x^2dx

∫√(u)*(1/3)du

= 2/3(u)^3/2 *(1/3)

= 2/9(x^3 + 1)^3/2

Is this the correct answer? Thank you.

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- Thread starter Lo.Lee.Ta.
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In summary, the conversation discusses the process of solving the integral of x^2 times the square root of x^3 + 1. The individual breaks down the steps involved, using u-substitution, and provides the final answer of 2/9(x^3 + 1)^3/2 with a reminder to include the constant of integration. There is also a reminder to always differentiate, even when confident in the answer, to ensure accuracy.

- #1

Lo.Lee.Ta.

- 217

- 0

u = x^3 + 1

du = 3x^2dx

(1/3)du = x^2dx

∫√(u)*(1/3)du

= 2/3(u)^3/2 *(1/3)

= 2/9(x^3 + 1)^3/2

Is this the correct answer? Thank you.

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- #2

SithsNGiggles

- 186

- 0

Yep, looks like it. You can always differentiate your answer if you're not sure.

- #3

Mentallic

Homework Helper

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Lo.Lee.Ta. said:

u = x^3 + 1

du = 3x^2dx

(1/3)du = x^2dx

∫√(u)*(1/3)du

= 2/3(u)^3/2 *(1/3)

= 2/9(x^3 + 1)^3/2

Is this the correct answer? Thank you.

Yes, but don't forget about the constant of integration.

- #4

Lo.Lee.Ta.

- 217

- 0

Thanks, you guys! :)

That's true. I should differentiate when I'm not sure...

That's true. I should differentiate when I'm not sure...

- #5

Ray Vickson

Science Advisor

Homework Helper

Dearly Missed

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Lo.Lee.Ta. said:Thanks, you guys! :)

That's true. I should differentiate when I'm not sure...

You should differentiate even when you are sure! (Sometimes a person can be sure but still have the wrong answer.)

In other words: always differentiate!

The integral of (x^2)(sqrt(x^3 + 1))dx represents the area under the curve of the function (x^2)(sqrt(x^3 + 1)) between the limits of integration.

To solve the integral of (x^2)(sqrt(x^3 + 1))dx, you can use the substitution method by setting u = x^3 + 1 and du = 3x^2 dx. This will transform the integral into ∫ (1/3)(sqrt(u)) du, which can be easily solved using the power rule and basic integration techniques.

Yes, the integral of (x^2)(sqrt(x^3 + 1))dx can be solved using the power rule after applying the substitution method to simplify the integral.

The limits of integration for the integral of (x^2)(sqrt(x^3 + 1))dx depend on the specific problem or scenario given. However, they typically involve the values of x that define the region under the curve of the function.

The integral of (x^2)(sqrt(x^3 + 1))dx has various real-world applications in fields such as physics, engineering, and economics. It can be used to calculate the volume of irregularly shaped objects, the work done by a variable force, and the area under a demand curve in economics, among others.

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