# integrate this by parts and substitution

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• Nov 6th 2012, 10:46 PM
99.95
integrate this by parts and substitution
imgur: the simple image sharer
for the first one,
i tried:

let e^x = u
du/dx = e^x
but then i couldn't put that back in (du = e^x dx is of no use)

then I tried let
e^2x = u

du = 2e^2x dx
putting it back in i had

1/2 Integral of [ ucos(u^1/2)du ]

let f = u let dg/du = cos(u^1/2)
df/du = 1 g = ?? Got ugly here i think.
Any ideas?
• Nov 6th 2012, 10:51 PM
chiro
Re: integrate this by parts and substitution
Hey 99.95.

Use the hint that e^x*e^x = e^2x where du = e^xdx and u = e^x.
• Nov 6th 2012, 10:53 PM
MarkFL
Re: integrate this by parts and substitution
Your first substitution attempt would give you:

$u=e^x\,\therefore\,du=e^x\,dx=u\,dx\,\therefore\,d x=\frac{du}{u}$ and your integral becomes:

$\int u\cos(u)\,du$

Now try integration by parts.
• Nov 6th 2012, 11:05 PM
99.95
Re: integrate this by parts and substitution
Thanks guys, got it!

Could I get some help for the second one? I'm quite bewildered
• Nov 6th 2012, 11:09 PM
MarkFL
Re: integrate this by parts and substitution
Let:

$u=\sqrt{x}\,\therefore\,du=\frac{1}{2\sqrt{x}}\,dx \,\therefore\,dx=2u\,du$ and the integral becomes:

$2\int ue^u\,du$

Now use integration by parts.
• Nov 6th 2012, 11:10 PM
chiro
Re: integrate this by parts and substitution
Try introducing a SQRT(x)*SQRT(x) term and just check that this is OK for the integral (it should be, but you'll need to check for yourself just in case). So instead of finding e^(SQRT(x))dx find SQRT(x)e^(SQRT(x))/SQRT(x)dx (Also this should work because you only have a single point where this breaks down which means it should be OK).