Calculus Examples

$\frac{e^{2 x}}{9 + e^{2 x}}$

Step 1

Let $u_{2} = 9 + e^{2 x}$ . Then $d u_{2} = 2 e^{2 x} d x$ , so $\frac{1}{2} d u_{2} = e^{2 x} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = 9 + e^{2 x}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $9 + e^{2 x}$ .

$\frac{d}{d x} [9 + e^{2 x}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $9 + e^{2 x}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [e^{2 x}]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [e^{2 x}]$

Step 1.1.2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [e^{2 x}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [e^{2 x}]$ .

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Step 1.1.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 1.1.3.1.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$0 + \frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [2 x]$

Step 1.1.3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$0 + e^{u_{1}} \frac{d}{d x} [2 x]$

Step 1.1.3.1.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$0 + e^{2 x} \frac{d}{d x} [2 x]$

Step 1.1.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$0 + e^{2 x} (2 \frac{d}{d x} [x])$

Step 1.1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0 + e^{2 x} (2 \cdot 1)$

Step 1.1.3.4

Multiply $2$ by $1$ .

$0 + e^{2 x} \cdot 2$

Step 1.1.3.5

Move $2$ to the left of $e^{2 x}$ .

$0 + 2 e^{2 x}$

Step 1.1.4

Add $0$ and $2 e^{2 x}$ .

$2 e^{2 x}$

Step 1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{2}$ .

$\int \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 2.2

Move $2$ to the left of $u_{2}$ .

$\int \frac{1}{2 u_{2}} d u_{2}$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{1}{u_{2}} d u_{2}$

Step 4

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{1}{2} (\ln (| u_{2} |) + C)$

Step 5

Simplify.

$\frac{1}{2} \ln (| u_{2} |) + C$

Step 6

Replace all occurrences of $u_{2}$ with $9 + e^{2 x}$ .

$\frac{1}{2} \ln (| 9 + e^{2 x} |) + C$