Integral of e^xsinxDate: 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/ |
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