Question

integral question! integral e^x sinx dx

Answer 1

\[\int\limits_{}^{}e ^{x}\sin(x)dx\]

Answer 2

Integrate by parts.
\[\int e^x\sin x~dx=uv-\int v~du\]
where you let
\[\begin{matrix}u=e^x&&&dv=\sin x~dx\\
du=e^x~dx&&&v=-\cos x\end{matrix}\]
\[\int e^x\sin x~dx=-e^x\cos x+\int e^x\cos x~dx\]
Integrate by parts again, this time letting
\[\begin{matrix}u=e^x&&&dv=\cos x~dx\\
du=e^x~dx&&&v=\sin x\end{matrix}\]
\[\int e^x\sin x~dx=-e^x\cos x+\left[e^x\sin x-\int e^x\sin x~dx\right]\\
2\int e^x\sin x~dx=e^x(\sin x-\cos x)\\
\int e^x\sin x~dx=\frac{e^x(\sin x-\cos x)}{2}\]

Answer 3

ahh it's that last bit i didn't get! setting it equal to the original equation! i was like... man this is gonna go on forever!!