Calculus Examples

$\frac{{(x + 2)}^{2}}{x^{4}}$

Step 1

Write $\frac{{(x + 2)}^{2}}{x^{4}}$ as a function.

$f (x) = \frac{{(x + 2)}^{2}}{x^{4}}$

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.

$F (x) = \int \frac{{(x + 2)}^{2}}{x^{4}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 4.2.1

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

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

Step 4.2.2

Multiply $4$ by $- 1$ .

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

Step 5

Let $u_{1} = x + 2$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 5.1

Let $u_{1} = x + 2$ . Find $\frac{d u_{1}}{d x}$ .

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Step 5.1.1

Differentiate $x + 2$ .

$\frac{d}{d x} [x + 2]$

Step 5.1.2

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [2]$

Step 5.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int {u_{1}}^{2} {(u_{1} - 2)}^{- 4} d u_{1}$

Step 6

Let $u_{2} = u_{1} - 2$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.1

Let $u_{2} = u_{1} - 2$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 6.1.1

Differentiate $u_{1} - 2$ .

$\frac{d}{d u_{1}} [u_{1} - 2]$

Step 6.1.2

By the Sum Rule, the derivative of $u_{1} - 2$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [- 2]$ .

$\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [- 2]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d u_{1}} [- 2]$

Step 6.1.4

Since $- 2$ is constant with respect to $u_{1}$ , the derivative of $- 2$ with respect to $u_{1}$ is $0$ .

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int {(u_{2} + 2)}^{2} {u_{2}}^{- 4} d u_{2}$

Step 7

Let $u_{3} = {u_{2}}^{2}$ . Then $d u_{3} = 2 u_{2} d u_{2}$ , so $\frac{1}{2} d u_{3} = u_{2} d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 7.1

Let $u_{3} = {u_{2}}^{2}$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 7.1.1

Differentiate ${u_{2}}^{2}$ .

$\frac{d}{d u_{2}} [{u_{2}}^{2}]$

Step 7.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$2 u_{2}$

Step 7.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\int {(\sqrt{u_{3}} + 2)}^{2} {\sqrt{u_{3}}}^{- 5} \frac{1}{2} d u_{3}$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{2}$ and ${(\sqrt{u_{3}} + 2)}^{2}$ .

$\int \frac{{(\sqrt{u_{3}} + 2)}^{2}}{2} {\sqrt{u_{3}}}^{- 5} d u_{3}$

Step 8.2

Combine $\frac{{(\sqrt{u_{3}} + 2)}^{2}}{2}$ and ${\sqrt{u_{3}}}^{- 5}$ .

$\int \frac{{(\sqrt{u_{3}} + 2)}^{2} {\sqrt{u_{3}}}^{- 5}}{2} d u_{3}$

Step 8.3

Move ${\sqrt{u_{3}}}^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int \frac{{(\sqrt{u_{3}} + 2)}^{2}}{2 {\sqrt{u_{3}}}^{5}} d u_{3}$

Step 8.4

Rewrite ${\sqrt{u_{3}}}^{5}$ as $\sqrt{{u_{3}}^{5}}$ .

$\int \frac{{(\sqrt{u_{3}} + 2)}^{2}}{2 \sqrt{{u_{3}}^{5}}} d u_{3}$

Step 9

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{{(\sqrt{u_{3}} + 2)}^{2}}{\sqrt{{u_{3}}^{5}}} d u_{3}$

Step 10

Apply basic rules of exponents.

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Step 10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u_{3}}$ as ${u_{3}}^{\frac{1}{2}}$ .

$\frac{1}{2} \int \frac{{({u_{3}}^{\frac{1}{2}} + 2)}^{2}}{\sqrt{{u_{3}}^{5}}} d u_{3}$

Step 10.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{u_{3}}^{5}}$ as ${u_{3}}^{\frac{5}{2}}$ .

$\frac{1}{2} \int \frac{{({u_{3}}^{\frac{1}{2}} + 2)}^{2}}{{u_{3}}^{\frac{5}{2}}} d u_{3}$

Step 10.3

Move ${u_{3}}^{\frac{5}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{2} \int {({u_{3}}^{\frac{1}{2}} + 2)}^{2} {({u_{3}}^{\frac{5}{2}})}^{- 1} d u_{3}$

Step 10.4

Multiply the exponents in ${({u_{3}}^{\frac{5}{2}})}^{- 1}$ .

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Step 10.4.1

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

$\frac{1}{2} \int {({u_{3}}^{\frac{1}{2}} + 2)}^{2} {u_{3}}^{\frac{5}{2} \cdot - 1} d u_{3}$

Step 10.4.2

Multiply $\frac{5}{2} \cdot - 1$ .

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Step 10.4.2.1

Combine $\frac{5}{2}$ and $- 1$ .

$\frac{1}{2} \int {({u_{3}}^{\frac{1}{2}} + 2)}^{2} {u_{3}}^{\frac{5 \cdot - 1}{2}} d u_{3}$

Step 10.4.2.2

Multiply $5$ by $- 1$ .

$\frac{1}{2} \int {({u_{3}}^{\frac{1}{2}} + 2)}^{2} {u_{3}}^{\frac{- 5}{2}} d u_{3}$

Step 10.4.3

Move the negative in front of the fraction.

$\frac{1}{2} \int {({u_{3}}^{\frac{1}{2}} + 2)}^{2} {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11

Simplify.

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Step 11.1

Rewrite ${({u_{3}}^{\frac{1}{2}} + 2)}^{2}$ as $({u_{3}}^{\frac{1}{2}} + 2) ({u_{3}}^{\frac{1}{2}} + 2)$ .

$\frac{1}{2} \int ({u_{3}}^{\frac{1}{2}} + 2) ({u_{3}}^{\frac{1}{2}} + 2) {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.2

Apply the distributive property.

$\frac{1}{2} \int ({u_{3}}^{\frac{1}{2}} ({u_{3}}^{\frac{1}{2}} + 2) + 2 ({u_{3}}^{\frac{1}{2}} + 2)) {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.3

Apply the distributive property.

$\frac{1}{2} \int ({u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} + {u_{3}}^{\frac{1}{2}} \cdot 2 + 2 ({u_{3}}^{\frac{1}{2}} + 2)) {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.4

Apply the distributive property.

$\frac{1}{2} \int ({u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} + {u_{3}}^{\frac{1}{2}} \cdot 2 + 2 {u_{3}}^{\frac{1}{2}} + 2 \cdot 2) {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.5

Apply the distributive property.

$\frac{1}{2} \int ({u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} + {u_{3}}^{\frac{1}{2}} \cdot 2) {u_{3}}^{- \frac{5}{2}} + (2 {u_{3}}^{\frac{1}{2}} + 2 \cdot 2) {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.6

Apply the distributive property.

$\frac{1}{2} \int {u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + {u_{3}}^{\frac{1}{2}} \cdot 2 {u_{3}}^{- \frac{5}{2}} + (2 {u_{3}}^{\frac{1}{2}} + 2 \cdot 2) {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.7

Apply the distributive property.

$\frac{1}{2} \int {u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + {u_{3}}^{\frac{1}{2}} \cdot 2 {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.8

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

$\frac{1}{2} \int {u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.9

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

$\frac{1}{2} \int {u_{3}}^{\frac{1}{2} + \frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.10

Combine the numerators over the common denominator.

$\frac{1}{2} \int {u_{3}}^{\frac{1 + 1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.11

Add $1$ and $1$ .

$\frac{1}{2} \int {u_{3}}^{\frac{2}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.12

Cancel the common factor of $2$ .

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Step 11.12.1

Cancel the common factor.

Step 11.12.2

Rewrite the expression.

$\frac{1}{2} \int {u_{3}}^{1} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.13

Simplify.

$\frac{1}{2} \int u_{3} \cdot {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.14

Raise $u_{3}$ to the power of $1$ .

$\frac{1}{2} \int {u_{3}}^{1} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.15

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

$\frac{1}{2} \int {u_{3}}^{1 - \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.16

Write $1$ as a fraction with a common denominator.

$\frac{1}{2} \int {u_{3}}^{\frac{2}{2} - \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.17

Combine the numerators over the common denominator.

$\frac{1}{2} \int {u_{3}}^{\frac{2 - 5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.18

Subtract $5$ from $2$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.19

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

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{\frac{1}{2} - \frac{5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.20

Combine the numerators over the common denominator.

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{\frac{1 - 5}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.21

Subtract $5$ from $1$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{\frac{- 4}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.22

Cancel the common factor of $- 4$ and $2$ .

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Step 11.22.1

Factor $2$ out of $- 4$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{\frac{2 \cdot - 2}{2}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.22.2

Cancel the common factors.

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Step 11.22.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{\frac{2 \cdot - 2}{2 (1)}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.22.2.2

Cancel the common factor.

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{\frac{2 \cdot - 2}{2 \cdot 1}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.22.2.3

Rewrite the expression.

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{\frac{- 2}{1}} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.22.2.4

Divide $- 2$ by $1$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.23

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

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{\frac{1}{2} - \frac{5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.24

Combine the numerators over the common denominator.

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{\frac{1 - 5}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.25

Subtract $5$ from $1$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{\frac{- 4}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.26

Cancel the common factor of $- 4$ and $2$ .

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Step 11.26.1

Factor $2$ out of $- 4$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{\frac{2 \cdot - 2}{2}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.26.2

Cancel the common factors.

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Step 11.26.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{\frac{2 \cdot - 2}{2 (1)}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.26.2.2

Cancel the common factor.

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{\frac{2 \cdot - 2}{2 \cdot 1}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.26.2.3

Rewrite the expression.

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{\frac{- 2}{1}} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.26.2.4

Divide $- 2$ by $1$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{- 2} + 2 \cdot 2 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.27

Multiply $2$ by $2$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 2 {u_{3}}^{- 2} + 2 {u_{3}}^{- 2} + 4 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.28

Add $2 {u_{3}}^{- 2}$ and $2 {u_{3}}^{- 2}$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 3}{2}} + 4 {u_{3}}^{- 2} + 4 {u_{3}}^{- \frac{5}{2}} d u_{3}$

Step 11.29

Move ${u_{3}}^{\frac{- 3}{2}}$ .

$\frac{1}{2} \int 4 {u_{3}}^{- 2} + 4 {u_{3}}^{- \frac{5}{2}} + {u_{3}}^{\frac{- 3}{2}} d u_{3}$

Step 12

Move the negative in front of the fraction.

$\frac{1}{2} \int 4 {u_{3}}^{- 2} + 4 {u_{3}}^{- \frac{5}{2}} + {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 13

Split the single integral into multiple integrals.

$\frac{1}{2} (\int 4 {u_{3}}^{- 2} d u_{3} + \int 4 {u_{3}}^{- \frac{5}{2}} d u_{3} + \int {u_{3}}^{- \frac{3}{2}} d u_{3})$

Step 14

Since $4$ is constant with respect to $u_{3}$ , move $4$ out of the integral.

$\frac{1}{2} (4 \int {u_{3}}^{- 2} d u_{3} + \int 4 {u_{3}}^{- \frac{5}{2}} d u_{3} + \int {u_{3}}^{- \frac{3}{2}} d u_{3})$

Step 15

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

$\frac{1}{2} (4 (- {u_{3}}^{- 1} + C) + \int 4 {u_{3}}^{- \frac{5}{2}} d u_{3} + \int {u_{3}}^{- \frac{3}{2}} d u_{3})$

Step 16

Since $4$ is constant with respect to $u_{3}$ , move $4$ out of the integral.

$\frac{1}{2} (4 (- {u_{3}}^{- 1} + C) + 4 \int {u_{3}}^{- \frac{5}{2}} d u_{3} + \int {u_{3}}^{- \frac{3}{2}} d u_{3})$

Step 17

By the Power Rule, the integral of ${u_{3}}^{- \frac{5}{2}}$ with respect to $u_{3}$ is $- \frac{2}{3} {u_{3}}^{- \frac{3}{2}}$ .

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

Step 18

Simplify.

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Step 18.1

Combine ${u_{3}}^{- \frac{3}{2}}$ and $\frac{2}{3}$ .

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

Step 18.2

Move $2$ to the left of ${u_{3}}^{- \frac{3}{2}}$ .

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

Step 18.3

Move ${u_{3}}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 19

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

$\frac{1}{2} (4 (- {u_{3}}^{- 1} + C) + 4 (- \frac{2}{3 {u_{3}}^{\frac{3}{2}}} + C) - 2 {u_{3}}^{- \frac{1}{2}} + C)$

Step 20

Simplify.

$\frac{1}{2} (- \frac{4}{u_{3}} - \frac{8}{3 {u_{3}}^{\frac{3}{2}}} - \frac{2}{{u_{3}}^{\frac{1}{2}}}) + C$

Step 21

Substitute back in for each integration substitution variable.

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Step 21.1

Replace all occurrences of $u_{3}$ with ${u_{2}}^{2}$ .

$\frac{1}{2} (- \frac{4}{{u_{2}}^{2}} - \frac{8}{3 {({u_{2}}^{2})}^{\frac{3}{2}}} - \frac{2}{{({u_{2}}^{2})}^{\frac{1}{2}}}) + C$

Step 21.2

Replace all occurrences of $u_{2}$ with $u_{1} - 2$ .

$\frac{1}{2} (- \frac{4}{{(u_{1} - 2)}^{2}} - \frac{8}{3 {({(u_{1} - 2)}^{2})}^{\frac{3}{2}}} - \frac{2}{{({(u_{1} - 2)}^{2})}^{\frac{1}{2}}}) + C$

Step 21.3

Replace all occurrences of $u_{1}$ with $x + 2$ .

$\frac{1}{2} (- \frac{4}{{(x + 2 - 2)}^{2}} - \frac{8}{3 {({(x + 2 - 2)}^{2})}^{\frac{3}{2}}} - \frac{2}{{({(x + 2 - 2)}^{2})}^{\frac{1}{2}}}) + C$

Step 22

Simplify.

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Step 22.1

Combine the opposite terms in $- \frac{4}{{(x + 2 - 2)}^{2}} - \frac{8}{3 {({(x + 2 - 2)}^{2})}^{\frac{3}{2}}} - \frac{2}{{({(x + 2 - 2)}^{2})}^{\frac{1}{2}}}$ .

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Step 22.1.1

Subtract $2$ from $2$ .

$\frac{1}{2} (- \frac{4}{{(x + 0)}^{2}} - \frac{8}{3 {({(x + 2 - 2)}^{2})}^{\frac{3}{2}}} - \frac{2}{{({(x + 2 - 2)}^{2})}^{\frac{1}{2}}}) + C$

Step 22.1.2

Add $x$ and $0$ .

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 {({(x + 2 - 2)}^{2})}^{\frac{3}{2}}} - \frac{2}{{({(x + 2 - 2)}^{2})}^{\frac{1}{2}}}) + C$

Step 22.1.3

Subtract $2$ from $2$ .

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 {({(x + 0)}^{2})}^{\frac{3}{2}}} - \frac{2}{{({(x + 2 - 2)}^{2})}^{\frac{1}{2}}}) + C$

Step 22.1.4

Add $x$ and $0$ .

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 {(x^{2})}^{\frac{3}{2}}} - \frac{2}{{({(x + 2 - 2)}^{2})}^{\frac{1}{2}}}) + C$

Step 22.1.5

Subtract $2$ from $2$ .

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 {(x^{2})}^{\frac{3}{2}}} - \frac{2}{{({(x + 0)}^{2})}^{\frac{1}{2}}}) + C$

Step 22.1.6

Add $x$ and $0$ .

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 {(x^{2})}^{\frac{3}{2}}} - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C$

Step 22.2

Simplify each term.

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Step 22.2.1

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

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Step 22.2.1.1

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

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 x^{2 (\frac{3}{2})}} - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C$

Step 22.2.1.2

Cancel the common factor of $2$ .

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Step 22.2.1.2.1

Cancel the common factor.

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 x^{2 (\frac{3}{2})}} - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C$

Step 22.2.1.2.2

Rewrite the expression.

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 x^{3}} - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C$

Step 22.2.2

Simplify the denominator.

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Step 22.2.2.1

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

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Step 22.2.2.1.1

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

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 x^{3}} - \frac{2}{x^{2 (\frac{1}{2})}}) + C$

Step 22.2.2.1.2

Cancel the common factor of $2$ .

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Step 22.2.2.1.2.1

Cancel the common factor.

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 x^{3}} - \frac{2}{x^{2 (\frac{1}{2})}}) + C$

Step 22.2.2.1.2.2

Rewrite the expression.

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 x^{3}} - \frac{2}{x^{1}}) + C$

Step 22.2.2.2

Simplify.

$\frac{1}{2} (- \frac{4}{x^{2}} - \frac{8}{3 x^{3}} - \frac{2}{x}) + C$

Step 22.3

Apply the distributive property.

$\frac{1}{2} (- \frac{4}{x^{2}}) + \frac{1}{2} (- \frac{8}{3 x^{3}}) + \frac{1}{2} (- \frac{2}{x}) + C$

Step 22.4

Simplify.

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Step 22.4.1

Cancel the common factor of $2$ .

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Step 22.4.1.1

Move the leading negative in $- \frac{4}{x^{2}}$ into the numerator.

$\frac{1}{2} \cdot \frac{- 4}{x^{2}} + \frac{1}{2} (- \frac{8}{3 x^{3}}) + \frac{1}{2} (- \frac{2}{x}) + C$

Step 22.4.1.2

Factor $2$ out of $- 4$ .

$\frac{1}{2} \cdot \frac{2 (- 2)}{x^{2}} + \frac{1}{2} (- \frac{8}{3 x^{3}}) + \frac{1}{2} (- \frac{2}{x}) + C$

Step 22.4.1.3

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot - 2}{x^{2}} + \frac{1}{2} (- \frac{8}{3 x^{3}}) + \frac{1}{2} (- \frac{2}{x}) + C$

Step 22.4.1.4

Rewrite the expression.

$\frac{- 2}{x^{2}} + \frac{1}{2} (- \frac{8}{3 x^{3}}) + \frac{1}{2} (- \frac{2}{x}) + C$

Step 22.4.2

Cancel the common factor of $2$ .

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Step 22.4.2.1

Move the leading negative in $- \frac{8}{3 x^{3}}$ into the numerator.

$\frac{- 2}{x^{2}} + \frac{1}{2} \cdot \frac{- 8}{3 x^{3}} + \frac{1}{2} (- \frac{2}{x}) + C$

Step 22.4.2.2

Factor $2$ out of $- 8$ .

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

Step 22.4.2.3

Cancel the common factor.

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

Step 22.4.2.4

Rewrite the expression.

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

Step 22.4.3

Cancel the common factor of $2$ .

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Step 22.4.3.1

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

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

Step 22.4.3.2

Factor $2$ out of $- 2$ .

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

Step 22.4.3.3

Cancel the common factor.

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

Step 22.4.3.4

Rewrite the expression.

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

Step 22.5

Simplify each term.

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Step 22.5.1

Move the negative in front of the fraction.

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

Step 22.5.2

Move the negative in front of the fraction.

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

Step 22.5.3

Move the negative in front of the fraction.

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

Step 23

The answer is the antiderivative of the function $f (x) = \frac{{(x + 2)}^{2}}{x^{4}}$ .

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