Question

How do you integrate `int sqrtx e^x dx` using integration by parts?

Answer 1

`sqrtxe^x-(sqrtpierfi(sqrtx))/2+C`

Explanation:

Problem:
`intsqrtxe^xdx`

Integrate by parts: `intfgprime=fg-intfprimeg`
`f=sqrtx`, `gprime=e^x`
`darr`
`fprime=1/(2sqrtx),g=e^x:`
=`sqrtxe^x-inte^x/(2sqrtx)dx`

Now solving:
`inte^x/(2sqrtx)dx`
Substitute `u=sqrtx->dx=2sqrtxdu:`
=`sqrtpi/2int(2e^[u^2])/sqrtpidu`

Now solving:
`int(2e^[u^2])/sqrtpidu`

This is a special integral, imaginary error function:
`=erfi(u)`

Plug in solved integrals:
`sqrtpi/2int(2e^[u^2])/sqrtpidu`
=`[sqrtpierfi(u)]/2`

Undo substitution `u=sqrtx:`
=`[sqrtpierfi(sqrtx)]/2`

Plug in solved integrals:
`sqrtxe^x-inte^x/2sqrtxdx`
=`sqrtxe^x-[sqrtpierfi(sqrtx)]/2`

The problem is solved:
`intsqrtxe^xdx`
=`sqrtxe^x-(sqrtpierfi(sqrtx))/2+C`