Calculus Examples

$\frac{e^{- \sqrt{x}}}{\sqrt{x}}$

Step 1

Write $\frac{e^{- \sqrt{x}}}{\sqrt{x}}$ as a function.

$f (x) = \frac{e^{- \sqrt{x}}}{\sqrt{x}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{e^{- \sqrt{x}}}{\sqrt{x}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$\int \frac{e^{- x^{\frac{1}{2}}}}{\sqrt{x}} d x$

Step 4.2

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

$\int \frac{e^{- x^{\frac{1}{2}}}}{x^{\frac{1}{2}}} d x$

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

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

$\int e^{- x^{\frac{1}{2}}} x^{\frac{- 1}{2}} d x$

Step 4.4.3

Move the negative in front of the fraction.

$\int e^{- x^{\frac{1}{2}}} x^{- \frac{1}{2}} d x$

Step 5

Let $u = - x^{\frac{1}{2}}$ . Then $d u = - \frac{1}{2 x^{\frac{1}{2}}} d x$ , so $1 d u = - \frac{1}{2 x^{\frac{1}{2}}} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = - x^{\frac{1}{2}}$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $- x^{\frac{1}{2}}$ .

$\frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 5.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$- \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$- (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 5.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 5.1.5

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

$- (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 5.1.6

Combine the numerators over the common denominator.

$- (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 5.1.7

Simplify the numerator.

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Step 5.1.7.1

Multiply $- 1$ by $2$ .

$- (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 5.1.7.2

Subtract $2$ from $1$ .

$- (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 5.1.8

Move the negative in front of the fraction.

$- (\frac{1}{2} x^{- \frac{1}{2}})$

Step 5.1.9

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

$- \frac{x^{- \frac{1}{2}}}{2}$

Step 5.1.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{2 x^{\frac{1}{2}}}$

Step 5.2

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

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

Step 6

Multiply $- 1$ by $2$ .

$\int - 2 e^{u} d u$

Step 7

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

$- 2 \int e^{u} d u$

Step 8

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$- 2 (e^{u} + C)$

Step 9

Simplify.

$- 2 e^{u} + C$

Step 10

Replace all occurrences of $u$ with $- x^{\frac{1}{2}}$ .

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

Step 11

The answer is the antiderivative of the function $f (x) = \frac{e^{- \sqrt{x}}}{\sqrt{x}}$ .

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