Calculus Examples

$\int \frac{3 e^{x}}{4 + e^{x}} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \frac{e^{x}}{4 + e^{x}} d x$

Step 2

Let $u = 4 + e^{x}$ . 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 2.1

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

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Step 2.1.1

Differentiate $4 + e^{x}$ .

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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 2.1.2.2

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 2.1.3

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 2.1.4

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

$e^{x}$

Step 2.2

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

$3 \int \frac{1}{u} d u$

Step 3

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$3 (\ln (| u |) + C)$

Step 4

Simplify.

$3 \ln (| u |) + C$

Step 5

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

$3 \ln (| 4 + e^{x} |) + C$