Question

What is `int_(0)^(2) x^3*e^(x^2)dx`?

Answer 1

`1/2(3e^4+ 1)`

Explanation:

First do substitution

`u = x^2`
`du = 2x dx`

Now we have

`1/2int_0^2u*e^u du`

Now by part

`v = u`
`v' = 1`
`w' = e^u`
`w = e^u`

if `x = 2 => u = 4`
if `x = 0 =>u = 0`

`1/2int_0^4u*e^u du = 1/2([u*e^u]_0^4-int_0^4e^udu)`

Which is

`1/2int_0^2u*e^u du = 1/2([u*e^u]_0^4-[e^u]_0^4)`

`1/2(3e^4+ 1)`