Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

$\int x e^{x} d x + \int - y e^{x} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

$x e^{x} - \int e^{x} d x + \int - y e^{x} d x$

Step 3

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$x e^{x} - (e^{x} + C) + \int - y e^{x} d x$

Step 4

Since $- y$ is constant with respect to $x$ , move $- y$ out of the integral.

$x e^{x} - (e^{x} + C) - y \int e^{x} d x$

Step 5

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$x e^{x} - (e^{x} + C) - y (e^{x} + C)$

Step 6

Simplify.

$x e^{x} - e^{x} - y e^{x} + C$