Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

$\int x \cdot x + x \frac{1}{x} + \frac{1}{x} x + \frac{1}{x} \cdot \frac{1}{x} d x$

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 $x$ .

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

Step 4.3.1.2

Cancel the common factor of $x$ .

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Step 4.3.1.2.1

Cancel the common factor.

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

Step 4.3.1.2.2

Rewrite the expression.

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

Step 4.3.1.3

Cancel the common factor of $x$ .

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Step 4.3.1.3.1

Cancel the common factor.

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

Step 4.3.1.3.2

Rewrite the expression.

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

Step 4.3.1.4

Multiply $\frac{1}{x} \cdot \frac{1}{x}$ .

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Step 4.3.1.4.1

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

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

Step 4.3.1.4.2

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

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

Step 4.3.1.4.3

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

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

Step 4.3.1.4.4

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

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

Step 4.3.1.4.5

Add $1$ and $1$ .

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

Step 4.3.2

Add $1$ and $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Apply basic rules of exponents.

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Step 8.1

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

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Multiply $2$ by $- 1$ .

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

Step 9

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

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

Step 10

Simplify.

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

Step 11

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

$F (x) =$ $\frac{1}{3} x^{3} + 2 x - x^{- 1} + C$