Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} - x$ .

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Step 4.1.2.1

Multiply by $1$ .

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

Step 4.1.2.2

Factor $x^{\frac{1}{2}}$ out of $- x$ .

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

Step 4.1.2.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- x^{\frac{1}{2}})$ .

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

Combine the numerators over the common denominator.

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

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $2$ by $2$ .

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

Step 4.3.5.2

Subtract $1$ from $4$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $3$ by $- 1$ .

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

Step 6.3

Move the negative in front of the fraction.

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

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

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

Step 7.3

Factor out negative.

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

Step 7.4

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

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

Step 7.5

Combine the numerators over the common denominator.

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

Step 7.6

Subtract $3$ from $1$ .

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

Step 7.7

Cancel the common factor of $- 2$ and $2$ .

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Step 7.7.1

Factor $2$ out of $- 2$ .

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

Step 7.7.2

Cancel the common factors.

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Step 7.7.2.1

Factor $2$ out of $2$ .

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

Step 7.7.2.2

Cancel the common factor.

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

Step 7.7.2.3

Rewrite the expression.

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

Step 7.7.2.4

Divide $- 1$ by $1$ .

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

Step 7.8

Reorder $x^{- \frac{3}{2}}$ and $- x^{- 1}$ .

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 10

The integral of $x^{- 1}$ with respect to $x$ is $\ln (| x |)$ .

$- (\ln (| x |) + C) + \int x^{- \frac{3}{2}} d x$

Step 11

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

$- (\ln (| x |) + C) - 2 x^{- \frac{1}{2}} + C$

Step 12

Simplify.

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Step 12.1

Simplify.

$- \ln (| x |) - 2 \frac{1}{x^{\frac{1}{2}}} + C$

Step 12.2

Simplify.

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Step 12.2.1

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

$- \ln (| x |) + \frac{- 2}{x^{\frac{1}{2}}} + C$

Step 12.2.2

Move the negative in front of the fraction.

$- \ln (| x |) - \frac{2}{x^{\frac{1}{2}}} + C$

Step 13

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

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