integrate x*e^x — step-by-step solution

Let's compute the integral:

\int(x e^{x})dx

For

x e^{x}

, we use integration by parts:

\int(u1 \cdot v2')dx = u1 \cdot v2 + \int(u1' \cdot v2)dx

, with u1 =

x

and v2' =

e^{x}

u1'

is equal to

1

Let's integrate v2':

\int(e^{x})dx

e^{x}

Let's compute the integral

u1 \cdot v2 + \int(u1' \cdot v2)dx

x e^{x} - \int(e^{x})dx

x e^{x} - e^{x}

You can factorize this expression into:

e^{x} \cdot (x - 1)

Indefinite integrals are defined up to an additive constant C, so this yields:

e^{x} \cdot (x - 1) + C

▷Result: $e^{x} \cdot (x - 1) + C$ , where C is a constant