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

Let's compute the integral:

\int x e^{x} d x

⇥For

x e^{x}

, we use integration by parts:

\int u1 \cdot v2' d x = u1 \cdot v2 - \int u1' \cdot v2 d x

, with u1 =

x

and v2' =

e^{x}

⇥Thus,

u1'

1

⇥Let's integrate v2':

\int e^{x} d x

⇥ⓘ The derivative of

e^{x}

e^{x}

, so

e^{x}

is an antiderivative of

e^{x}

\int e^{x} d x

e^{x}

⇥

e^{x}

⇥Let's calculate

u1 \cdot v2 - \int u1' \cdot v2 d x

x e^{x} - \int e^{x} d x

⇥ⓘ The derivative of

- e^{x}

- e^{x}

, so

- e^{x}

is an antiderivative of

- e^{x}

- \int e^{x} d x

- e^{x}

⇥

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

Then you can resume the integral computation:

ⅰ

\int x e^{x} d x

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

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

This expression can be factorized 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