Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Apply basic rules of exponents.

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Step 4.1

Apply the product rule to $\frac{1 - x}{x}$ .

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Multiply $2$ by $- 1$ .

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

Step 5

Let $u_{1} = 1 - x$ . Then $d u_{1} = - d x$ , so $- d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 5.1

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

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Step 5.1.1

Differentiate $1 - x$ .

$\frac{d}{d x} [1 - x]$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

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

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

Step 5.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- x]$ .

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Step 5.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$0 - \frac{d}{d x} [x]$

Step 5.1.3.2

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

$0 - 1 \cdot 1$

Step 5.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 5.1.4

Subtract $1$ from $0$ .

$- 1$

Step 5.2

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $- u_{1} + 1$ .

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

Step 7.1.2

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

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

Step 7.1.3

Evaluate $\frac{d}{d u_{1}} [- u_{1}]$ .

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Step 7.1.3.1

Since $- 1$ is constant with respect to $u_{1}$ , the derivative of $- u_{1}$ with respect to $u_{1}$ is $- \frac{d}{d u_{1}} [u_{1}]$ .

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

Step 7.1.3.2

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

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

Step 7.1.3.3

Multiply $- 1$ by $1$ .

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

Step 7.1.4

Differentiate using the Constant Rule.

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Step 7.1.4.1

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

$- 1 + 0$

Step 7.1.4.2

Add $- 1$ and $0$ .

$- 1$

Step 7.2

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Multiply $- 1$ by $- 1$ .

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

Step 9.2

Multiply $\int {(- u_{2} + 1)}^{2} {u_{2}}^{- 2} d u_{2}$ by $1$ .

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

Step 10

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 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

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 10.2

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

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

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

Step 12

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

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

Step 13

Apply basic rules of exponents.

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

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

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Step 13.4.1

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

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

Step 13.4.2

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

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Step 13.4.2.1

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

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

Step 13.4.2.2

Multiply $3$ by $- 1$ .

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

Step 13.4.3

Move the negative in front of the fraction.

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

Step 14

Simplify.

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Step 14.1

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

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

Step 14.2

Apply the distributive property.

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

Step 14.3

Apply the distributive property.

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

Step 14.4

Apply the distributive property.

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

Step 14.5

Apply the distributive property.

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

Step 14.6

Apply the distributive property.

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

Step 14.7

Apply the distributive property.

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

Step 14.8

Move ${u_{3}}^{\frac{1}{2}}$ .

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

Step 14.9

Move ${u_{3}}^{\frac{1}{2}}$ .

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

Step 14.10

Multiply $- 1$ by $- 1$ .

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

Step 14.11

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

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

Step 14.12

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

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

Step 14.13

Combine the numerators over the common denominator.

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

Step 14.14

Add $1$ and $1$ .

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

Step 14.15

Cancel the common factor of $2$ .

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Step 14.15.1

Cancel the common factor.

Step 14.15.2

Rewrite the expression.

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

Step 14.16

Simplify.

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

Step 14.17

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

Step 14.18

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

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

Step 14.19

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

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

Step 14.20

Combine the numerators over the common denominator.

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

Step 14.21

Subtract $3$ from $2$ .

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

Step 14.22

Multiply $- 1$ by $1$ .

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

Step 14.23

Factor out negative.

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

Step 14.24

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

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

Step 14.25

Combine the numerators over the common denominator.

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

Step 14.26

Subtract $3$ from $1$ .

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

Step 14.27

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

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Step 14.27.1

Factor $2$ out of $- 2$ .

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

Step 14.27.2

Cancel the common factors.

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Step 14.27.2.1

Factor $2$ out of $2$ .

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

Step 14.27.2.2

Cancel the common factor.

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

Step 14.27.2.3

Rewrite the expression.

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

Step 14.27.2.4

Divide $- 1$ by $1$ .

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

Step 14.28

Multiply $- 1$ by $1$ .

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

Step 14.29

Factor out negative.

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

Step 14.30

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

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

Step 14.31

Combine the numerators over the common denominator.

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

Step 14.32

Subtract $3$ from $1$ .

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

Step 14.33

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

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Step 14.33.1

Factor $2$ out of $- 2$ .

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

Step 14.33.2

Cancel the common factors.

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Step 14.33.2.1

Factor $2$ out of $2$ .

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

Step 14.33.2.2

Cancel the common factor.

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

Step 14.33.2.3

Rewrite the expression.

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

Step 14.33.2.4

Divide $- 1$ by $1$ .

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

Step 14.34

Multiply $1$ by $1$ .

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

Step 14.35

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

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

Step 14.36

Subtract ${u_{3}}^{- 1}$ from $- {u_{3}}^{- 1}$ .

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

Step 14.37

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

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

Step 15

Move the negative in front of the fraction.

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

Step 16

Split the single integral into multiple integrals.

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

Step 17

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

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

Step 18

The integral of ${u_{3}}^{- 1}$ with respect to $u_{3}$ is $\ln (| u_{3} |)$ .

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

Step 19

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

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

Step 20

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

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

Step 21

Simplify.

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

Step 22

Substitute back in for each integration substitution variable.

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Step 22.1

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

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

Step 22.2

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

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

Step 22.3

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

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

Step 23

Simplify.

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Step 23.1

Simplify each term.

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Step 23.1.1

Simplify each term.

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Step 23.1.1.1

Apply the distributive property.

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

Step 23.1.1.2

Multiply $- 1$ by $1$ .

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

Step 23.1.1.3

Multiply $- - x$ .

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Step 23.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 23.1.1.3.2

Multiply $x$ by $1$ .

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

Step 23.1.2

Combine the opposite terms in $- 1 + x + 1$ .

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Step 23.1.2.1

Add $- 1$ and $1$ .

$\frac{1}{2} (- 2 \ln (| {(x + 0)}^{2} |) + 2 {({(- (1 - x) + 1)}^{2})}^{\frac{1}{2}} - \frac{2}{{({(- (1 - x) + 1)}^{2})}^{\frac{1}{2}}}) + C$

Step 23.1.2.2

Add $x$ and $0$ .

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

Step 23.1.3

Remove non-negative terms from the absolute value.

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Step 23.1.3.1

Add $- 1$ and $1$ .

$\frac{1}{2} (- 2 \ln ({(x + 0)}^{2}) + 2 {({(- (1 - x) + 1)}^{2})}^{\frac{1}{2}} - \frac{2}{{({(- (1 - x) + 1)}^{2})}^{\frac{1}{2}}}) + C$

Step 23.1.3.2

Add $x$ and $0$ .

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

Step 23.1.4

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

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Step 23.1.4.1

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

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

Step 23.1.4.2

Cancel the common factor of $2$ .

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Step 23.1.4.2.1

Cancel the common factor.

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

Step 23.1.4.2.2

Rewrite the expression.

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

Step 23.1.5

Simplify each term.

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Step 23.1.5.1

Apply the distributive property.

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

Step 23.1.5.2

Multiply $- 1$ by $1$ .

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

Step 23.1.5.3

Multiply $- - x$ .

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Step 23.1.5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 23.1.5.3.2

Multiply $x$ by $1$ .

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

Step 23.1.6

Combine the opposite terms in $- 1 + x + 1$ .

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Step 23.1.6.1

Add $- 1$ and $1$ .

$\frac{1}{2} (- 2 \ln (x^{2}) + 2 {(x + 0)}^{1} - \frac{2}{{({(- (1 - x) + 1)}^{2})}^{\frac{1}{2}}}) + C$

Step 23.1.6.2

Add $x$ and $0$ .

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

Step 23.1.7

Simplify.

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

Step 23.1.8

Simplify the denominator.

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Step 23.1.8.1

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

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Step 23.1.8.1.1

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

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

Step 23.1.8.1.2

Cancel the common factor of $2$ .

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Step 23.1.8.1.2.1

Cancel the common factor.

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

Step 23.1.8.1.2.2

Rewrite the expression.

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

Step 23.1.8.2

Simplify each term.

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Step 23.1.8.2.1

Apply the distributive property.

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

Step 23.1.8.2.2

Multiply $- 1$ by $1$ .

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

Step 23.1.8.2.3

Multiply $- - x$ .

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Step 23.1.8.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 23.1.8.2.3.2

Multiply $x$ by $1$ .

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

Step 23.1.8.3

Combine the opposite terms in $- 1 + x + 1$ .

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Step 23.1.8.3.1

Add $- 1$ and $1$ .

$\frac{1}{2} (- 2 \ln (x^{2}) + 2 x - \frac{2}{{(x + 0)}^{1}}) + C$

Step 23.1.8.3.2

Add $x$ and $0$ .

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

Step 23.1.8.4

Simplify.

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

Step 23.2

Apply the distributive property.

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

Step 23.3

Simplify.

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Step 23.3.1

Cancel the common factor of $2$ .

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Step 23.3.1.1

Factor $2$ out of $- 2 \ln (x^{2})$ .

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

Step 23.3.1.2

Cancel the common factor.

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

Step 23.3.1.3

Rewrite the expression.

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

Step 23.3.2

Cancel the common factor of $2$ .

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Step 23.3.2.1

Factor $2$ out of $2 x$ .

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

Step 23.3.2.2

Cancel the common factor.

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

Step 23.3.2.3

Rewrite the expression.

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

Step 23.3.3

Cancel the common factor of $2$ .

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Step 23.3.3.1

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

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

Step 23.3.3.2

Factor $2$ out of $- 2$ .

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

Step 23.3.3.3

Cancel the common factor.

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

Step 23.3.3.4

Rewrite the expression.

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

Step 23.4

Move the negative in front of the fraction.

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

Step 24

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

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