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Evaluate the following integral

I think this is a recursion type problem, but I'm not quite sure. Again, I could be going about this problem horribly. Just need someone to check to see if this is the correct answer or if I'm even close to doing this problem right.

\(\displaystyle \int sin(x) e^x \, dx\)

\(\displaystyle u = sinx\)

\(\displaystyle du = cosx dx\)

\(\displaystyle dv = e^x\)

\(\displaystyle v = e^x\)

\(\displaystyle sinx e^x - \int e^x cosx \,dx\)

\(\displaystyle u = e^x \)

\(\displaystyle du = e^x \)

\(\displaystyle dv = -sinx\)

\(\displaystyle v = cosx\)

\(\displaystyle sinx e^x - e^x cos x - \int cosx e^x dx\)

\(\displaystyle u = cosx\)

\(\displaystyle du = -sinx\)

\(\displaystyle dv = e^x\)

\(\displaystyle v = e^x\)

\(\displaystyle I = sinx e^x - cosx e^x - cosx e ^x + \int e^x sinx \, dx\)

\(\displaystyle 2I = sinx e^x - 2cosx e^x\)

\(\displaystyle I = \frac{sinx e^x - 2cosx e^x}{2}\)

I think this is a recursion type problem, but I'm not quite sure. Again, I could be going about this problem horribly. Just need someone to check to see if this is the correct answer or if I'm even close to doing this problem right.

\(\displaystyle \int sin(x) e^x \, dx\)

\(\displaystyle u = sinx\)

\(\displaystyle du = cosx dx\)

\(\displaystyle dv = e^x\)

\(\displaystyle v = e^x\)

\(\displaystyle sinx e^x - \int e^x cosx \,dx\)

\(\displaystyle u = e^x \)

\(\displaystyle du = e^x \)

\(\displaystyle dv = -sinx\)

\(\displaystyle v = cosx\)

\(\displaystyle sinx e^x - e^x cos x - \int cosx e^x dx\)

\(\displaystyle u = cosx\)

\(\displaystyle du = -sinx\)

\(\displaystyle dv = e^x\)

\(\displaystyle v = e^x\)

\(\displaystyle I = sinx e^x - cosx e^x - cosx e ^x + \int e^x sinx \, dx\)

\(\displaystyle 2I = sinx e^x - 2cosx e^x\)

\(\displaystyle I = \frac{sinx e^x - 2cosx e^x}{2}\)

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