Question

How do you find the definite integral of `x^2 * sqrt (x^3 + 1) * dx` in the interval `[1,2]`?

Answer 1

We wish to evaluate `int_1^2 x^2sqrt(x^3+1)dx`.

Let `u = x^3 + 1`. Then `du = 3x^2dx` and `dx = (du)/(3x^2)`. Change the bounds of integration.

`I = int_2^9 x^2sqrt(u) * (du)/(3x^2)`

`I = 1/3int_2^9 sqrt(u)`

`I = 1/3[2/3u^(3/2)]_2^9`

`I = 2/9(9)^(3/2) - 2/9(2)^(3/2) = 6 - 4/9sqrt(2) ~~ 5.37`

Hopefully this helps!