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Find

First we integrate the corresponding indefinite integral using integration by parts. We find that

Since

xe^{x} – e^{x} + C

is the family of all antiderivatives of xe^{x}, we can get one particular antiderivative by taking C = 0. Then we can use the resulting antiderivative

xe^{x} – e^{x}

in the FTC.

We take

u = x

v' = e^{x}

u' = 1

v = e^{x}.

Then we apply the formula, keeping the limits of integration:

Finally we evaluate each term from 0 to 1:

It's reassuring that we got the same answer we got with Way 1.

Make it rain.