Integral Evaluation for x^2√(2+x^3) Using Substitution or Power Method

[Kinetica]

Homework Statement

integral:

x² sqrt(2+x³) dx

Homework Equations

The Attempt at a Solution

Should I use substitution? Should I do it as a power?

 

[Bohrok]

U substitution is the way to go for this one.

 

[HallsofIvy]

Since the only difficulty is that "$2+ x^3$" inside the square root, let $u= 2+ x^3$ seems an obvious way to go.

 

[Kinetica]

So when I substitute it as HallsofIvy suggested, what do I do with x²dx?
x²dx=du?

integral:
sqrt(u)du=2u^3/2/3...

stuck.

 

[Dick]

If u=2+x^3 isn't du=3*x^2*dx? Isn't that how substitution works?

 

[Kinetica]

I am extremely confused with the next step of the solution.

integral
sqrt(u)du/(3x²) is this the correct substitution?

 

[Dick]

Well, no. What happened to the x^2 in your original problem? Isn't it x^2*sqrt(u)*(du/(3*x^2))?

 

[Kinetica]

Thanks goodness, I got it.