Integral of e^xsinx

Date: 01/23/2001 at 10:11:52
From: george
Subject: integral of e^xsinx

I have tried to intgrate e^xsinx by parts, but I always get e^xsinx or 
e^xcosx to integrate, and it just keeps going. Could you show me the 
steps to this indefinite integral?

Date: 01/23/2001 at 12:03:29
From: Doctor Rob
Subject: Re: Integral of e^xsinx

Thanks for writing to Ask Dr. Math, George.

There is a pretty clever trick to this one.  Integrate by parts twice
as follows:

   INTEGRAL u dv = u*v - INTEGRAL v du,
   u = sin(x), dv = e^x dx, du = cos(x) dx, v = e^x:

   INTEGRAL e^x*sin(x) dx = e^x*sin(x) - INTEGRAL e^x*cos(x) dx.

   INTEGRAL u dv = u*v - INTEGRAL v du,
   u = cos(x), dv = e^x dx, du = -sin(x) dx, v = e^x:

   INTEGRAL e^x*sin(x) dx
                = e^x*sin(x) - [e^x*cos(x) - INTEGRAL -e^x*sin(x) dx],
                = e^x*[sin(x)-cos(x)] - INTEGRAL e^x*sin(x) dx.

Now it may seem that you didn't get anywhere by doing this, but notice
that the integral on the left and the one on the right are the same.
Now just combine like terms by adding that term to both sides, and
finally solve for that integral to get your answer.

You can easily verify the answer by differentiation.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/