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A product with an exponential

CalculusMedium

Integrate $x e^{x}$ with respect to $x$ between the limits $0$ and $1$ . Report the result as an integer.

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Hint

One factor becomes simpler when differentiated, and the other does not change when integrated.

Solution

Integrate by parts with $u = x$ and $d v = e^{x} d x$ , so $d u = d x$ and $v = e^{x}$ .
Then an antiderivative of $x e^{x}$ is $x e^{x} - e^{x} = (x - 1) e^{x}$ .
At $x = 1$ this is $0$ , and at $x = 0$ it is $- 1$ .
So the integral equals $0 - (- 1) = 1$ .

Answer: $1$