Calculus Examples

$\frac{\sqrt{x}}{x}$

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

Split the fraction into multiple fractions.

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Step 1.2.1

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

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

Step 1.2.2

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.2.2.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 1.2.2.1.1

Raise $x$ to the power of $1$ .

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

Step 1.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 1.2.2.3

Combine the numerators over the common denominator.

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

Step 1.2.2.4

Subtract $1$ from $2$ .

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

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

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

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

Step 1.3.2

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

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Move the negative in front of the fraction.

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

Step 2

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

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