Question

How do you integrate `(x^2)(e^x)dx`?

Answer 1

`intx^2e^xdx=e^x(x^2-2x+2)+c`

Explanation:

We do it using integration by parts.

Let `u=x^2` and `v=e^x`, then `du=2xdx` and `dv=e^xdx`

Now integration by parts states that

`intu(x)v'(x)dx=u(x)v(x)-intv(x)u'(x)dx`

Hence `intx^2e^xdx=x^2e^x-inte^x xx 2xdx`

= `x^2e^x-2intxe^xdx+c` ...............(1)

Now we set `u=x`, then `du=dx`

and `intxe^xdx=xe^x-inte^x xx1xxdx` or

`intxe^xdx=xe^x-inte^xdx=xe^x-e^x`

Putting this in (1), we get

`intx^2e^xdx=x^2e^x-2(xe^x-e^x)+c`

= `e^x(x^2-2x+2)+c`