# integration by parts

• Mar 22nd 2009, 05:39 PM
qzno
integration by parts
use integration by parts to find:

$\int^{\frac{\pi}{2}}_{0} xsin^2xcosx dx$
• Mar 22nd 2009, 06:05 PM
skeeter
Quote:

Originally Posted by qzno
use integration by parts to find:

$\int^{\frac{\pi}{2}}_{0} xsin^2xcosx dx$

$u = x$

$du = dx$

$dv = \sin^2{x}\cos{x} \, dx$

$v = \frac{\sin^3{x}}{3}$

$\int x\sin^2{x}\cos{x} \, dx = \frac{x\sin^3{x}}{3} - \int \frac{\sin^3{x}}{3} \, dx$

for the last integral ...

$\sin^3{x} = (1 - \cos^2{x})\sin{x} = \sin{x} + \cos^2{x}(-\sin{x})$
• Mar 22nd 2009, 06:07 PM
qzno
$dv = \sin^2{x}\cos{x} \, dx$

$v = \frac{\sin^3{x}}{3}$

is this a rule?
• Mar 22nd 2009, 06:22 PM
skeeter
Quote:

Originally Posted by qzno
$dv = \sin^2{x}\cos{x} \, dx$

$v = \frac{\sin^3{x}}{3}$

is this a rule?

it's an antiderivative using informal substitution ... note that $\cos{x}$ is the derivative of $\sin{x}$ ...

$\sin^2{x}\cos{x} \, dx$ is the same as $t^2 dt$