Calculus Examples

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

Step 1

Write ${(x - \frac{1}{2 x})}^{2}$ as a function.

$f (x) = {(x - \frac{1}{2 x})}^{2}$

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.

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

Step 4

Simplify.

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Step 4.1

Rewrite ${(x - \frac{1}{2 x})}^{2}$ as $(x - \frac{1}{2 x}) (x - \frac{1}{2 x})$ .

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

Step 4.2

Expand $(x - \frac{1}{2 x}) (x - \frac{1}{2 x})$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $x$ by $x$ .

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

Step 4.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.3

Cancel the common factor of $x$ .

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Step 4.3.1.3.1

Factor $x$ out of $- x$ .

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

Step 4.3.1.3.2

Factor $x$ out of $2 x$ .

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

Step 4.3.1.3.3

Cancel the common factor.

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

Step 4.3.1.3.4

Rewrite the expression.

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

Step 4.3.1.4

Cancel the common factor of $x$ .

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Step 4.3.1.4.1

Move the leading negative in $- \frac{1}{2 x}$ into the numerator.

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

Step 4.3.1.4.2

Factor $x$ out of $2 x$ .

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

Step 4.3.1.4.3

Cancel the common factor.

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

Step 4.3.1.4.4

Rewrite the expression.

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

Step 4.3.1.5

Move the negative in front of the fraction.

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

Step 4.3.1.6

Multiply $- \frac{1}{2 x} (- \frac{1}{2 x})$ .

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Step 4.3.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.6.2

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

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

Step 4.3.1.6.3

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

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

Step 4.3.1.6.4

Multiply $2$ by $2$ .

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

Step 4.3.1.6.5

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

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

Step 4.3.1.6.6

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

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

Step 4.3.1.6.7

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

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

Step 4.3.1.6.8

Add $1$ and $1$ .

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

Step 4.3.2

Subtract $\frac{1}{2}$ from $- \frac{1}{2}$ .

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Factor $2$ out of $- 2$ .

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

$\frac{1}{3} x^{3} + C + \int - 1 d x + \int \frac{1}{4 x^{2}} d x$

Step 7

Apply the constant rule.

$\frac{1}{3} x^{3} + C - x + C + \int \frac{1}{4 x^{2}} d x$

Step 8

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{3} x^{3} + C - x + C + \frac{1}{4} \int \frac{1}{x^{2}} d x$

Step 9

Apply basic rules of exponents.

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Step 9.1

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

$\frac{1}{3} x^{3} + C - x + C + \frac{1}{4} \int {(x^{2})}^{- 1} d x$

Step 9.2

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

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Step 9.2.1

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

$\frac{1}{3} x^{3} + C - x + C + \frac{1}{4} \int x^{2 \cdot - 1} d x$

Step 9.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{3} x^{3} + C - x + C + \frac{1}{4} \int x^{- 2} d x$

Step 10

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

$\frac{1}{3} x^{3} + C - x + C + \frac{1}{4} (- x^{- 1} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{1}{3} x^{3} - x + \frac{1}{4} (- x^{- 1}) + C$

Step 11.2

Simplify.

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Step 11.2.1

Combine $\frac{1}{4}$ and $x^{- 1}$ .

$\frac{1}{3} x^{3} - x - \frac{x^{- 1}}{4} + C$

Step 11.2.2

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

$\frac{1}{3} x^{3} - x - \frac{1}{4 x} + C$

Step 12

The answer is the antiderivative of the function $f (x) = {(x - \frac{1}{2 x})}^{2}$ .

$F (x) =$ $\frac{1}{3} x^{3} - x - \frac{1}{4 x} + C$