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?

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.

Re: integrate this by parts and substitution

Your first substitution attempt would give you:

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

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

Now try integration by parts.

Re: integrate this by parts and substitution

Thanks guys, got it!

Could I get some help for the second one? I'm quite bewildered

Re: integrate this by parts and substitution

Let:

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

$\displaystyle 2\int ue^u\,du$

Now use integration by parts.

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).