Problems on Techniques of Integration

Use the Integration by Parts for definite integrals. Set

$$\left\{\begin{array}{lll} u &=&x\\ dv &=& e^{x}dx\;. \end{array}\right.$$

Then

$$\left\{\begin{array}{lll} du &=& dx\\ v &=& e^{x}\;. \end{array}\right.$$

So

$$\int_0^1 xe^{x} dx = [x e^{x}]_0^1 - \int_0^1 e^{x} dx$$

or

$$\int_0^1 x e^{x} dx = [x e^{x}]_0^1 - [e^{x}]_0^1 = e - (e-1) = 1\;.$$

Detailed Answer.

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