Question

How do you integrate `int 2x^2sqrt(x^3+1)dx` from [1,2]?

Answer 1

`12 - 8/9sqrt(2)`

Explanation:

This is a substitution problem. Let `u =x^3 + 1`. Then `du = 3x^2dx` and `dx = (du)/(3x^2)`.

`int_1^2 2x^2sqrt(u) * (du)/(3x^2)`

`int_1^2 2/3sqrt(u) du`

`2/3int_1^2 sqrt(u)du`

`2/3[2/3u^(3/2)]_1^2`

`2/3[2/3(x^3 + 1)^(3/2)]_1^2`

`2/3[2/3(2^3 + 1)^(3/2) - 2/3(1^3 +1)^(3/2)]`

`2/3[2/3(27) - 2/3sqrt(8)]`

`2/3[18 - 4/3sqrt(2)]`

`12 - 8/9sqrt(2)`

Hopefully this helps!