Calculus Examples

$\int \frac{{(4 + e^{x})}^{2}}{e^{x}} d x$

Step 1

Let $u = 4 + e^{x}$ . Then $d u = e^{x} d x$ , so $- d u = - e^{x} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 + e^{x}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 + e^{x}$ .

$\frac{d}{d x} [4 + e^{x}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $4 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [e^{x}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [e^{x}]$

Step 1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [e^{x}]$

Step 1.1.3

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

$0 + e^{x}$

Step 1.1.4

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

$e^{x}$

Step 1.2

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

$\int \frac{1}{1} u^{2} d u$

Step 2

Simplify.

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Step 2.1

Cancel the common factor of $1$ .

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Step 2.1.1

Cancel the common factor.

$\int \frac{1}{1} u^{2} d u$

Step 2.1.2

Rewrite the expression.

$\int 1 u^{2} d u$

Step 2.2

Multiply $u^{2}$ by $1$ .

$\int u^{2} d u$

Step 3

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 4

Replace all occurrences of $u$ with $4 + e^{x}$ .

$\frac{1}{3} {(4 + e^{x})}^{3} + C$