Calculus Examples

$\int \frac{e^{x}}{\sqrt{2 + e^{x}}} d x$

Step 1

Let $u = 2 + e^{x}$ . 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 1.1

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

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Step 1.1.1

Differentiate $2 + e^{x}$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ 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 $d u$ .

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 2.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

$2 u^{\frac{1}{2}} + C$

Step 4

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

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