Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Simplify.

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

Step 5.2

Simplify.

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Step 5.2.1

Add $- e^{x}$ and $e^{x}$ .

$x e^{x} + 0 + C$

Step 5.2.2

Add $x e^{x}$ and $0$ .

$x e^{x} + C$