# integration by parts

• Mar 22nd 2009, 04:39 PM
qzno
integration by parts
use integration by parts to find:

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

Originally Posted by qzno
use integration by parts to find:

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

$\displaystyle u = x$

$\displaystyle du = dx$

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

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

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

for the last integral ...

$\displaystyle \sin^3{x} = (1 - \cos^2{x})\sin{x} = \sin{x} + \cos^2{x}(-\sin{x})$
• Mar 22nd 2009, 05:07 PM
qzno
$\displaystyle dv = \sin^2{x}\cos{x} \, dx$

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

is this a rule?
• Mar 22nd 2009, 05:22 PM
skeeter
Quote:

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

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

is this a rule?

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

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