Question

How do you integrate `(x^5)*(e^(x^2/2))`?

Answer 1

`(4x^4-4x^2-8)e^(x^2/2)+c`

Explanation:

Let's assume `x^2/2 = t`
`=> (2xdx)/2 = t, x dx = dt`

`=int(x^5)*(e^(x^2/2))dx` can be written as,

`=int(x^2)(x^2)x(e^(x^2/2))dx`

Substituting with `t` we get

`=int(2t)(2t)(e^t)dt`

`=4intt^2(e^t)dt`

Using Integration by Parts,

`∫(I)(II)dx=(I)∫(II)dx−∫((I)'∫(II)dx)dx`

where `(I)` and `(II)` are functions of `x`, and `(I)` represents which will be differentiated and `(II)` will be integrated subsequently in the above formula

Similarly following for the problem,

`=4(t^2)inte^tdt-4int((t^2)'inte^tdt)dt+c`

`=4t^2e^t-4int2te^tdt+c`

`=4t^2e^t-8intte^tdt+c`

Applying again integration by parts in second term, we get

`=4t^2e^t-8(te^t-inte^tdt)+c`

`=4t^2e^t-8(te^t-e^t)+c`

`=4t^2e^t-8te^t-8e^t+c`

Substituting `t` back,

`=4(x^2/2)^2e^(x^2/2)-8(x^2/2)e^(x^2/2)-8e^(x^2/2)+c`, where `c` is a constant

`=(4x^4-4x^2-8)e^(x^2/2)+c`