Calculus Examples

$\int e^{x} {(e^{x} + 1)}^{2} d x$

Step 1

Let $u = e^{x} + 1$ . Then $d u = e^{x} d x$ , so $\frac{1}{e^{x}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = e^{x} + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $e^{x} + 1$ .

$\frac{d}{d x} [e^{x} + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $e^{x} + 1$ with respect to $x$ is $\frac{d}{d x} [e^{x}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [e^{x}] + \frac{d}{d x} [1]$

Step 1.1.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x} + \frac{d}{d x} [1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$e^{x} + 0$

Step 1.1.4.2

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

$e^{x}$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int u^{2} d u$

Step 2

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{3} u^{3} + C$

Step 3

Replace all occurrences of $u$ with $e^{x} + 1$ .

$\frac{1}{3} {(e^{x} + 1)}^{3} + C$