How Do You Integrate x²(√2 + x) Using Substitution?

[romeIAM]

Homework Statement

∫x²(√2+x)

Homework Equations

∫f(x) from a to b = f'(b) - f'(a)
and substitution rule

The Attempt at a Solution

[/B]
I decided to make u=√(2+x), du= 1/2√(2+x) and when solving dx, i got dx= 2√(2+x) du. Substituting and then simplifying, I managed to get ∫2(x^2)u^2 du. But i can't go further from there. i can't find a way to get rid of the x², I didn't get far using u= x² or u=x+2 so I'm pretty sure I'm using the right substitution. I need help.

 

[Dick]

Homework Statement

∫x²(√2+x)

Homework Equations

∫f(x) from a to b = f'(b) - f'(a)
and substitution rule

The Attempt at a Solution

[/B]
I decided to make u=√(2+x), du= 1/2√(2+x) and when solving dx, i got dx= 2√(2+x) du. Substituting and then simplifying, I managed to get ∫2(x^2)u^2 du. But i can't go further from there. i can't find a way to get rid of the x², I didn't get far using u= x² or u=x+2 so I'm pretty sure I'm using the right substitution. I need help.

Try making u=2+x. Then x=u-2. So x^2=(u-2)^2. Take it from there.

 

[Mark44]

I managed to get ∫2(x^2)u^2 du.

When you do a substitution, do a complete substitution. In this case, neither x nor dx should appear after you make the substitution.

Also, in your original integral, you omitted dx. It's not a good habit to get into to ignore the differential. Doing so will come back to bite you in other integration techniques, including trig substitution and integration by parts.

 

[ehild]

Homework Statement

∫x²(√2+x)

The Attempt at a Solution

[/B]
I decided to make u=√(2+x), du= 1/2√(2+x) here. i can't find a way to get rid of the x², I didn't get far using u= x² or u=x+2 so I'm pretty sure I'm using the right substitution. I need help.

No need to substitute. Just expand the integrand, and integrate the sum by terms.