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What's the integral of x*e^x?

The integral of x*e^x is (x-1)*e^x + C, where C is the constant of integration.

To find the integral of x*e^x, we can use integration by parts. Let u = x and dv/dx = e^x. Then du/dx = 1 and v = e^x. Using the formula for integration by parts, we have:

∫x*e^x dx = x*e^x - ∫e^x dx

The integral of e^x is simply e^x, so we have:

∫x*e^x dx = x*e^x - e^x + C

where C is the constant of integration. We can simplify this expression by factoring out e^x:

∫x*e^x dx = (x-1)*e^x + C

This is our final answer. We can check it by differentiating to see if we get back to the original function:

d/dx [(x-1)*e^x] = e^x + x*e^x - e^x = x*e^x

So the derivative of (x-1)*e^x is indeed x*e^x, which confirms that our answer is correct.

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