Question

\[\int\limits_{}^{}e ^{x}sinx dx\]

Answer 1

\[\int\limits\limits_{}^{}e ^{x}sinx dx\]Integration by parts.\[u = e^x, dv = sinxdx\]\[du = e^x dx, v = -cosx\]\[\int\limits udv = uv - \int\limits vdu\]\[\int\limits e^x sinx dx = -e^x \cos x + \int\limits e^x \cos xdx\]\[u = e^x, dv = cosxdx\]\[du = e^x dx, v = \sin x\]\[\int\limits\limits e^x sinx dx = -e^x \cos x + e^x \sin x - \int\limits e^x sinx dx\]Bring the integral of e^x sinx to the left side.\[2\int\limits\limits\limits e^x sinx dx = -e^x \cos x + e^x \sin x \rightarrow\int\limits\limits\limits e^x sinx dx = \frac {1}{2}e^x(\sin x - \cos x) + C\]

Answer 2

is this a loopy example?

Answer 3

What do you mean? This one requires iterative use of parts. You could say it is slightly trickier than the average integration by parts.

Answer 4

oh ok never mind. thank you!

Answer 5

Your welcome =)