Calculus Examples

$\frac{\sqrt{x}}{x^{2}}$

Step 1

Write $\frac{\sqrt{x}}{x^{2}}$ as a function.

$f (x) = \frac{\sqrt{x}}{x^{2}}$

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.

$F (x) = \int \frac{\sqrt{x}}{x^{2}} d x$

Step 4

Simplify the expression.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

Multiply $2$ by $- 1$ .

$\int \sqrt{x} x^{- 2} d x$

Step 4.2.2

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

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

Step 4.2.3

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

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Step 4.2.3.1

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

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

Step 4.2.3.2

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

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

Step 4.2.3.3

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

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

Step 4.2.3.4

Combine the numerators over the common denominator.

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

Step 4.2.3.5

Simplify the numerator.

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Step 4.2.3.5.1

Multiply $- 2$ by $2$ .

$\int x^{\frac{1 - 4}{2}} d x$

Step 4.2.3.5.2

Subtract $4$ from $1$ .

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

Step 4.2.3.6

Move the negative in front of the fraction.

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

Step 5

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 6

Simplify the answer.

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Step 6.1

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 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

Move the negative in front of the fraction.

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

Step 7

The answer is the antiderivative of the function $f (x) = \frac{\sqrt{x}}{x^{2}}$ .

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