Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 4.3.1.2

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

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

Step 4.3.1.3

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

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

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 $\frac{1}{x}$ .

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

Step 4.3.1.4.2

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

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

Step 4.3.1.4.3

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

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

Step 4.3.1.4.4

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

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

Step 4.3.1.4.5

Add $1$ and $1$ .

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

Step 4.3.2

Add $\frac{1}{x}$ and $\frac{1}{x}$ .

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

Step 4.4

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 8

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

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

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.

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

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, ${(a^{m})}^{n} = a^{m n}$ .

$x + C + 2 (\ln (| x |) + C) + \int x^{2 \cdot - 1} d x$

Step 9.2.2

Multiply $2$ by $- 1$ .

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

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

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