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What's the integral of xe^xsin(x)?

The integral of xe^xsin(x) is (x-1/2)e^xsin(x) - 1/2cos(x)e^x + C.

To solve this integral, we will use integration by parts. Let u = x and dv = e^xsin(x)dx. Then du/dx = 1 and v = -1/2cos(x)e^x - 1/2sin(x)e^x. Using the formula for integration by parts, we have:

∫xe^xsin(x)dx = uv - ∫vdu
= x(-1/2cos(x)e^x - 1/2sin(x)e^x) - ∫(-1/2cos(x)e^x - 1/2sin(x)e^x)dx
= (x-1/2)e^xsin(x) - 1/2cos(x)e^x + C

Therefore, the integral of xe^xsin(x) is (x-1/2)e^xsin(x) - 1/2cos(x)e^x + C.

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