Question

How do you find the definite integral of `int x(x^(2/3))dx` from `[0,1]`?

Answer 1

`int_0^1 x(x^(2/3))d x=3/8`

Explanation:

`int_0^1 x(x^(2/3))d x=int_0^1 x root(3)(x^2)d x`
`int_0^1 x(x^(2/3))d x=int_0^1 root(3)(x^3*x^2)d x`
`int_0^1 x(x^(2/3))d x=int_0^1 x^(5/3) d x`
`int_0^1 x(x^(2/3))d x=|3/8*x^(8/3)|_0^1`
`int_0^1 x(x^(2/3))d x=3/8*1^(8/3)-3/8*0^(8/3)`
`int_0^1 x(x^(2/3))d x=3/8`