# reduction formula

• Feb 23rd 2009, 05:56 PM
jackm7
reduction formula
hey there, first time poster, long time struggler...so i need some help with a problem.

it says to find the reduction formula for the integral (e^x)(sin x)^n in terms of the integral (e^x)(sin x)^n-2.

i know its integration by parts but i have no idea what to do for my substitutions because (sin x)^n on its own is difficult enough so how do I incorporate the e^x?

cheers!
• Feb 24th 2009, 05:14 AM
Jester
Quote:

Originally Posted by jackm7
hey there, first time poster, long time struggler...so i need some help with a problem.

it says to find the reduction formula for the integral (e^x)(sin x)^n in terms of the integral (e^x)(sin x)^n-2.

i know its integration by parts but i have no idea what to do for my substitutions because (sin x)^n on its own is difficult enough so how do I incorporate the e^x?

cheers!

Integration by parts twice. For the first use

$\displaystyle u = \sin^n x,\; dv = e^x\,dx$

so

$\displaystyle du = n \sin ^{n-1}x \cos x\, dx \;\;v = e^x$

and

$\displaystyle e^x \sin^n x - n \int e^x \sin ^{n-1}x \cos x\,dx$

For the second by parts

$\displaystyle u = \sin ^{n-1}x \cos x,\; dv = e^x\,dx$

and use some identites - see how that goes.