Calculus Examples

$\frac{1}{x \sqrt{x}} d x$

Step 1

Write $\frac{1}{x \sqrt{x}} d x$ as a function.

$f (x) = \frac{1}{x \sqrt{x}} d x$

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.

$F (x) = \int \frac{1}{x \sqrt{x}} \cdot (d x) d x$

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

Tap for more steps...

Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.2

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Add $2$ and $1$ .

$\int \frac{1}{x^{\frac{3}{2}}} \cdot (d x) d x$

Step 4.3

Combine $d$ and $\frac{1}{x^{\frac{3}{2}}}$ .

$\int \frac{d}{x^{\frac{3}{2}}} \cdot x d x$

Step 4.4

Combine $\frac{d}{x^{\frac{3}{2}}}$ and $x$ .

$\int \frac{d x}{x^{\frac{3}{2}}} d x$

Step 4.5

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

$\int \frac{d}{x^{\frac{3}{2}} x^{- 1}} d x$

Step 4.6

Multiply $x^{\frac{3}{2}}$ by $x^{- 1}$ by adding the exponents.

Tap for more steps...

Step 4.6.1

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

$\int \frac{d}{x^{\frac{3}{2} - 1}} d x$

Step 4.6.2

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

$\int \frac{d}{x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}} d x$

Step 4.6.3

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

$\int \frac{d}{x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}} d x$

Step 4.6.4

Combine the numerators over the common denominator.

$\int \frac{d}{x^{\frac{3 - 1 \cdot 2}{2}}} d x$

Step 4.6.5

Simplify the numerator.

Tap for more steps...

Step 4.6.5.1

Multiply $- 1$ by $2$ .

$\int \frac{d}{x^{\frac{3 - 2}{2}}} d x$

Step 4.6.5.2

Subtract $2$ from $3$ .

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

Step 5

Since $d$ is constant with respect to $x$ , move $d$ out of the integral.

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

Step 6

Apply basic rules of exponents.

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

Tap for more steps...

Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Rewrite $d (2 x^{\frac{1}{2}} + C)$ as $d \cdot 2 x^{\frac{1}{2}} + C$ .

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

Step 8.2

Move $2$ to the left of $d$ .

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

Step 9

The answer is the antiderivative of the function $f (x) = \frac{1}{x \sqrt{x}} \cdot (d x)$ .

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