Integral of x·eˣ

\int x e^{x}\,dx = x e^{x} - e^{x} + C.

Differentiate it: the derivative of x e^{x} - e^{x} is x e^{x}. CalcViz checks every antiderivative this way before showing it.

\int x e^{x}\,dx = x e^{x} - e^{x} + C

Integration by parts with u = x (it simplifies when differentiated) and dv = eˣ dx.

Integration by parts
∫u dv = uv − ∫v du with u = $x$ (chosen by LIATE) and dv = $e^{x}\,dx$ , so du = $dx$ and v = $e^{x}$ .
$\int x e^{x}\,dx = x e^{x} - \int e^{x}\,dx$
Exponential rule
∫eᵘ du = eᵘ.
$\int e^{x}\,dx = e^{x}$

Constant C = 0.0

Every choice of C gives a valid antiderivative, since shifting up or down doesn’t change the slope.

Questions students ask

What is the integral of x·eˣ?

$\int x e^{x}\,dx = x e^{x} - e^{x} + C$ .

How do you integrate x·eˣ?

Integration by parts with u = x (it simplifies when differentiated) and dv = eˣ dx.

How can I check the answer?

Differentiate it: the derivative of $x e^{x} - e^{x}$ is $x e^{x}$ . CalcViz checks every antiderivative this way before showing it.