Integral of $x e^{x}$

Q: What is the integral of x * exp(x)?

The integral of x e^x is ≤ft(x - 1) e^x + C.

The integral of $x e^{x}$ is $\left(x - 1\right) e^{x} + C$ .

\int x e^{x}\,dx = \left(x - 1\right) e^{x} + C

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Graph of f(x) = x * exp(x) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 1.

How to work it out

1
We want an antiderivative of f(x) = x * exp(x).
$\int x e^{x}\,dx$
2
Method: integration by parts with u = x, dv = eˣ dx.
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int x e^{x}\,dx = \left(x - 1\right) e^{x} + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\left(x - 1\right) e^{x}\right] = e^{x} + \left(x - 1\right) e^{x}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} x e^{x}\,dx = \Big[\left(x - 1\right) e^{x}\Big]_{0}^{1} \approx 1$

LaTeX source of the formulas on this page

\[ \int x e^{x}\,dx = \left(x - 1\right) e^{x} + C \]
\[ \int x e^{x}\,dx \]
\[ \frac{d}{dx}\left[\left(x - 1\right) e^{x}\right] = e^{x} + \left(x - 1\right) e^{x} \]
\[ \int_{0}^{1} x e^{x}\,dx = \Big[\left(x - 1\right) e^{x}\Big]_{0}^{1} \approx 1 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	1
from 0 to 0.5	0.17563936

Questions

What is the integral of x * exp(x)?

The integral of $x e^{x}$ is $\left(x - 1\right) e^{x} + C$ .

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of x * exp(x) from 0 to 1?

About 1.

Integral of xexx e^{x}xex

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $x e^{x}$