Calculus Examples

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

Step 1

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 2

Set up the integral to solve.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify.

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Step 3.1.1

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

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

Step 3.1.2

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

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Step 3.1.2.1

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

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

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 3.1.2.2

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

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

Step 3.1.2.3

Combine the numerators over the common denominator.

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

Step 3.1.2.4

Add $2$ and $1$ .

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

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Move $x^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 3.2.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{- 1}$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.2.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 3.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.2.2.3

Move the negative in front of the fraction.

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

Step 4

By the Power Rule, the integral of $x^{- \frac{3}{2}}$ with respect to $x$ is $- 2 x^{- \frac{1}{2}}$ .

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

Step 5

Simplify the answer.

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Step 5.1

Rewrite $- 2 x^{- \frac{1}{2}} + C$ as $- 2 \frac{1}{x^{\frac{1}{2}}} + C$ .

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

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

Move the negative in front of the fraction.

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

Step 6

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

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