# Integration by subs. or parts?

• August 22nd 2006, 03:13 AM
sterps
Integration by subs. or parts?
I have this question in a quiz for uni:
http://users.bigpond.net.au/sterps/pictures/int.JPG

They show the exact same question in my lecture notes equalling:

1/2 e^x^2 + C , though im confused on how they get rid of the x in front: xe^x^2

any help would be appreciated
• August 22nd 2006, 03:29 AM
topsquark
Quote:

Originally Posted by sterps
I have this question in a quiz for uni:
http://users.bigpond.net.au/sterps/pictures/int.JPG

They show the exact same question in my lecture notes equalling:

1/2 e^x^2 + C , though im confused on how they get rid of the x in front: xe^x^2

any help would be appreciated

$\int dx \, xe^{x^2}$

Let $y = x^2$, then $dy = 2x dx$

Then
$\int dx \, xe^{x^2} = \int \frac{dy}{2}e^y$

$= \frac{1}{2}e^y + C = \frac{1}{2}e^{x^2} + C$

I don't know of a way to do this using integration by parts because $\int dx \, e^{x^2}$ can't be done exactly.

-Dan
• August 22nd 2006, 06:18 AM
CaptainBlack
You can do this without needing to integrate.

Just differentiate (a) through (d), one of these derivatives will be the
integrand, and that will be the correct solution.

RonL
• August 22nd 2006, 10:18 AM
Jameson
Quote:

Originally Posted by CaptainBlack
You can do this without needing to integrate.

Just differentiate (a) through (d), one of these derivatives will be the
integrand, and that will be the correct solution.

RonL

Yes, but that doesn't help integrating techniques much :)
• August 22nd 2006, 10:35 AM
CaptainBlack
Quote:

Originally Posted by Jameson
Yes, but that doesn't help integrating techniques much :)

It does if it teaches the student that they are allowed to spot the derivative
under the integral sign, or to use the fundamental theorem where they might
not have thought of it.

RonL
• August 23rd 2006, 05:20 AM
topsquark
And solving the problem backwards is always a good trick to have when doing multiple choice tests.

-Dan