Calculus Examples

$\int (x + 1) e^{x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x + 1$ and $d v = e^{x}$ .

$(x + 1) e^{x} - \int e^{x} d x$

Step 2

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

$(x + 1) e^{x} - (e^{x} + C)$

Step 3

Simplify the answer.

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Step 3.1

Rewrite $(x + 1) e^{x} - (e^{x} + C)$ as $x e^{x} + e^{x} - e^{x} + C$ .

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

Step 3.2

Simplify.

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Step 3.2.1

Subtract $e^{x}$ from $e^{x}$ .

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

Step 3.2.2

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

$x e^{x} + C$