Calculus Examples

$\frac{d y}{d x} = \frac{e^{x}}{5 + e^{x}}$

Step 1

Rewrite the equation.

$d y = \frac{e^{x}}{5 + e^{x}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{e^{x}}{5 + e^{x}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{e^{x}}{5 + e^{x}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 5 + e^{x}$ . 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 2.3.1.1

Let $u = 5 + e^{x}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $5 + e^{x}$ .

$\frac{d}{d x} [5 + e^{x}]$

Step 2.3.1.1.2

Differentiate.

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Step 2.3.1.1.2.1

By the Sum Rule, the derivative of $5 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [e^{x}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [e^{x}]$

Step 2.3.1.1.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

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

Step 2.3.1.1.3

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 2.3.1.1.4

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

$e^{x}$

Step 2.3.1.2

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

$y + C_{1} = \int \frac{1}{u} d u$

Step 2.3.2

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

$y + C_{1} = \ln (| u |) + C_{2}$

Step 2.3.3

Replace all occurrences of $u$ with $5 + e^{x}$ .

$y + C_{1} = \ln (| 5 + e^{x} |) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$y = \ln (| 5 + e^{x} |) + C$