Calculus Examples

$\int {(\frac{5 + r}{r})}^{2} d r$

Step 1

Apply basic rules of exponents.

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Step 1.1

Apply the product rule to $\frac{5 + r}{r}$ .

$\int \frac{{(5 + r)}^{2}}{r^{2}} d r$

Step 1.2

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

$\int {(5 + r)}^{2} {(r^{2})}^{- 1} d r$

Step 1.3

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

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Step 1.3.1

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

$\int {(5 + r)}^{2} r^{2 \cdot - 1} d r$

Step 1.3.2

Multiply $2$ by $- 1$ .

$\int {(5 + r)}^{2} r^{- 2} d r$

Step 2

Let $u_{1} = 5 + r$ . Then $d u_{1} = d r$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.1

Let $u_{1} = 5 + r$ . Find $\frac{d u_{1}}{d r}$ .

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Step 2.1.1

Differentiate $5 + r$ .

$\frac{d}{d r} [5 + r]$

Step 2.1.2

By the Sum Rule, the derivative of $5 + r$ with respect to $r$ is $\frac{d}{d r} [5] + \frac{d}{d r} [r]$ .

$\frac{d}{d r} [5] + \frac{d}{d r} [r]$

Step 2.1.3

Since $5$ is constant with respect to $r$ , the derivative of $5$ with respect to $r$ is $0$ .

$0 + \frac{d}{d r} [r]$

Step 2.1.4

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

$0 + 1$

Step 2.1.5

Add $0$ and $1$ .

$1$

Step 2.2

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

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

Step 3

Let $u_{2} = u_{1} - 5$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 3.1

Let $u_{2} = u_{1} - 5$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 3.1.1

Differentiate $u_{1} - 5$ .

$\frac{d}{d u_{1}} [u_{1} - 5]$

Step 3.1.2

By the Sum Rule, the derivative of $u_{1} - 5$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [- 5]$ .

$\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [- 5]$

Step 3.1.3

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

$1 + \frac{d}{d u_{1}} [- 5]$

Step 3.1.4

Since $- 5$ is constant with respect to $u_{1}$ , the derivative of $- 5$ with respect to $u_{1}$ is $0$ .

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

Step 3.2

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

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

Step 4

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 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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 4.2

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

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

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and ${(\sqrt{u_{3}} + 5)}^{2}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

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

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

Step 7

Apply basic rules of exponents.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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Step 7.4.1

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

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

Step 7.4.2

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

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Step 7.4.2.1

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

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

Step 7.4.2.2

Multiply $3$ by $- 1$ .

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

Step 7.4.3

Move the negative in front of the fraction.

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

Step 8

Simplify.

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Step 8.1

Rewrite ${({u_{3}}^{\frac{1}{2}} + 5)}^{2}$ as $({u_{3}}^{\frac{1}{2}} + 5) ({u_{3}}^{\frac{1}{2}} + 5)$ .

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Apply the distributive property.

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

Step 8.4

Apply the distributive property.

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

Step 8.5

Apply the distributive property.

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

Step 8.6

Apply the distributive property.

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

Step 8.7

Apply the distributive property.

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

Step 8.8

Reorder ${u_{3}}^{\frac{1}{2}}$ and $5$ .

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

Step 8.9

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

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

Step 8.10

Combine the numerators over the common denominator.

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

Step 8.11

Add $1$ and $1$ .

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

Step 8.12

Cancel the common factor of $2$ .

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Step 8.12.1

Cancel the common factor.

Step 8.12.2

Rewrite the expression.

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

Step 8.13

Simplify.

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

Step 8.14

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

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

Step 8.15

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

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

Step 8.16

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

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

Step 8.17

Combine the numerators over the common denominator.

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

Step 8.18

Subtract $3$ from $2$ .

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

Step 8.19

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

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

Step 8.20

Combine the numerators over the common denominator.

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

Step 8.21

Subtract $3$ from $1$ .

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

Step 8.22

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

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Step 8.22.1

Factor $2$ out of $- 2$ .

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

Step 8.22.2

Cancel the common factors.

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Step 8.22.2.1

Factor $2$ out of $2$ .

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

Step 8.22.2.2

Cancel the common factor.

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

Step 8.22.2.3

Rewrite the expression.

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

Step 8.22.2.4

Divide $- 1$ by $1$ .

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

Step 8.23

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

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

Step 8.24

Combine the numerators over the common denominator.

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

Step 8.25

Subtract $3$ from $1$ .

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

Step 8.26

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

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Step 8.26.1

Factor $2$ out of $- 2$ .

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

Step 8.26.2

Cancel the common factors.

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Step 8.26.2.1

Factor $2$ out of $2$ .

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

Step 8.26.2.2

Cancel the common factor.

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

Step 8.26.2.3

Rewrite the expression.

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

Step 8.26.2.4

Divide $- 1$ by $1$ .

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

Step 8.27

Multiply $5$ by $5$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} + 5 {u_{3}}^{- 1} + 5 {u_{3}}^{- 1} + 25 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 8.28

Add $5 {u_{3}}^{- 1}$ and $5 {u_{3}}^{- 1}$ .

$\frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} + 10 {u_{3}}^{- 1} + 25 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 8.29

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

$\frac{1}{2} \int 10 {u_{3}}^{- 1} + 25 {u_{3}}^{- \frac{3}{2}} + {u_{3}}^{\frac{- 1}{2}} d u_{3}$

Step 9

Move the negative in front of the fraction.

$\frac{1}{2} \int 10 {u_{3}}^{- 1} + 25 {u_{3}}^{- \frac{3}{2}} + {u_{3}}^{- \frac{1}{2}} d u_{3}$

Step 10

Split the single integral into multiple integrals.

$\frac{1}{2} (\int 10 {u_{3}}^{- 1} d u_{3} + \int 25 {u_{3}}^{- \frac{3}{2}} d u_{3} + \int {u_{3}}^{- \frac{1}{2}} d u_{3})$

Step 11

Since $10$ is constant with respect to $u_{3}$ , move $10$ out of the integral.

$\frac{1}{2} (10 \int {u_{3}}^{- 1} d u_{3} + \int 25 {u_{3}}^{- \frac{3}{2}} d u_{3} + \int {u_{3}}^{- \frac{1}{2}} d u_{3})$

Step 12

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

$\frac{1}{2} (10 (\ln (| u_{3} |) + C) + \int 25 {u_{3}}^{- \frac{3}{2}} d u_{3} + \int {u_{3}}^{- \frac{1}{2}} d u_{3})$

Step 13

Since $25$ is constant with respect to $u_{3}$ , move $25$ out of the integral.

$\frac{1}{2} (10 (\ln (| u_{3} |) + C) + 25 \int {u_{3}}^{- \frac{3}{2}} d u_{3} + \int {u_{3}}^{- \frac{1}{2}} d u_{3})$

Step 14

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

$\frac{1}{2} (10 (\ln (| u_{3} |) + C) + 25 (- 2 {u_{3}}^{- \frac{1}{2}} + C) + \int {u_{3}}^{- \frac{1}{2}} d u_{3})$

Step 15

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

$\frac{1}{2} (10 (\ln (| u_{3} |) + C) + 25 (- 2 {u_{3}}^{- \frac{1}{2}} + C) + 2 {u_{3}}^{\frac{1}{2}} + C)$

Step 16

Simplify.

$\frac{1}{2} (10 \ln (| u_{3} |) - \frac{50}{{u_{3}}^{\frac{1}{2}}} + 2 {u_{3}}^{\frac{1}{2}}) + C$

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

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

$\frac{1}{2} (10 \ln (| {u_{2}}^{2} |) - \frac{50}{{({u_{2}}^{2})}^{\frac{1}{2}}} + 2 {({u_{2}}^{2})}^{\frac{1}{2}}) + C$

Step 17.2

Replace all occurrences of $u_{2}$ with $u_{1} - 5$ .

$\frac{1}{2} (10 \ln (| {(u_{1} - 5)}^{2} |) - \frac{50}{{({(u_{1} - 5)}^{2})}^{\frac{1}{2}}} + 2 {({(u_{1} - 5)}^{2})}^{\frac{1}{2}}) + C$

Step 17.3

Replace all occurrences of $u_{1}$ with $5 + r$ .

$\frac{1}{2} (10 \ln (| {(5 + r - 5)}^{2} |) - \frac{50}{{({(5 + r - 5)}^{2})}^{\frac{1}{2}}} + 2 {({(5 + r - 5)}^{2})}^{\frac{1}{2}}) + C$

Step 18

Simplify.

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Step 18.1

Combine the opposite terms in $10 \ln (| {(5 + r - 5)}^{2} |) - \frac{50}{{({(5 + r - 5)}^{2})}^{\frac{1}{2}}} + 2 {({(5 + r - 5)}^{2})}^{\frac{1}{2}}$ .

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Step 18.1.1

Subtract $5$ from $5$ .

$\frac{1}{2} (10 \ln (| {(r + 0)}^{2} |) - \frac{50}{{({(5 + r - 5)}^{2})}^{\frac{1}{2}}} + 2 {({(5 + r - 5)}^{2})}^{\frac{1}{2}}) + C$

Step 18.1.2

Add $r$ and $0$ .

$\frac{1}{2} (10 \ln (| r^{2} |) - \frac{50}{{({(5 + r - 5)}^{2})}^{\frac{1}{2}}} + 2 {({(5 + r - 5)}^{2})}^{\frac{1}{2}}) + C$

Step 18.1.3

Subtract $5$ from $5$ .

$\frac{1}{2} (10 \ln (| r^{2} |) - \frac{50}{{({(r + 0)}^{2})}^{\frac{1}{2}}} + 2 {({(5 + r - 5)}^{2})}^{\frac{1}{2}}) + C$

Step 18.1.4

Add $r$ and $0$ .

$\frac{1}{2} (10 \ln (| r^{2} |) - \frac{50}{{(r^{2})}^{\frac{1}{2}}} + 2 {({(5 + r - 5)}^{2})}^{\frac{1}{2}}) + C$

Step 18.1.5

Subtract $5$ from $5$ .

$\frac{1}{2} (10 \ln (| r^{2} |) - \frac{50}{{(r^{2})}^{\frac{1}{2}}} + 2 {({(r + 0)}^{2})}^{\frac{1}{2}}) + C$

Step 18.1.6

Add $r$ and $0$ .

$\frac{1}{2} (10 \ln (| r^{2} |) - \frac{50}{{(r^{2})}^{\frac{1}{2}}} + 2 {(r^{2})}^{\frac{1}{2}}) + C$

Step 18.2

Simplify each term.

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Step 18.2.1

Remove non-negative terms from the absolute value.

$\frac{1}{2} (10 \ln (r^{2}) - \frac{50}{{(r^{2})}^{\frac{1}{2}}} + 2 {(r^{2})}^{\frac{1}{2}}) + C$

Step 18.2.2

Simplify the denominator.

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Step 18.2.2.1

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

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Step 18.2.2.1.1

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

$\frac{1}{2} (10 \ln (r^{2}) - \frac{50}{r^{2 (\frac{1}{2})}} + 2 {(r^{2})}^{\frac{1}{2}}) + C$

Step 18.2.2.1.2

Cancel the common factor of $2$ .

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Step 18.2.2.1.2.1

Cancel the common factor.

$\frac{1}{2} (10 \ln (r^{2}) - \frac{50}{r^{2 (\frac{1}{2})}} + 2 {(r^{2})}^{\frac{1}{2}}) + C$

Step 18.2.2.1.2.2

Rewrite the expression.

$\frac{1}{2} (10 \ln (r^{2}) - \frac{50}{r^{1}} + 2 {(r^{2})}^{\frac{1}{2}}) + C$

Step 18.2.2.2

Simplify.

$\frac{1}{2} (10 \ln (r^{2}) - \frac{50}{r} + 2 {(r^{2})}^{\frac{1}{2}}) + C$

Step 18.2.3

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

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Step 18.2.3.1

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

$\frac{1}{2} (10 \ln (r^{2}) - \frac{50}{r} + 2 r^{2 (\frac{1}{2})}) + C$

Step 18.2.3.2

Cancel the common factor of $2$ .

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Step 18.2.3.2.1

Cancel the common factor.

$\frac{1}{2} (10 \ln (r^{2}) - \frac{50}{r} + 2 r^{2 (\frac{1}{2})}) + C$

Step 18.2.3.2.2

Rewrite the expression.

$\frac{1}{2} (10 \ln (r^{2}) - \frac{50}{r} + 2 r^{1}) + C$

Step 18.2.4

Simplify.

$\frac{1}{2} (10 \ln (r^{2}) - \frac{50}{r} + 2 r) + C$

Step 18.3

Apply the distributive property.

$\frac{1}{2} (10 \ln (r^{2})) + \frac{1}{2} (- \frac{50}{r}) + \frac{1}{2} (2 r) + C$

Step 18.4

Simplify.

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Step 18.4.1

Cancel the common factor of $2$ .

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Step 18.4.1.1

Factor $2$ out of $10 \ln (r^{2})$ .

$\frac{1}{2} (2 (5 \ln (r^{2}))) + \frac{1}{2} (- \frac{50}{r}) + \frac{1}{2} (2 r) + C$

Step 18.4.1.2

Cancel the common factor.

$\frac{1}{2} (2 (5 \ln (r^{2}))) + \frac{1}{2} (- \frac{50}{r}) + \frac{1}{2} (2 r) + C$

Step 18.4.1.3

Rewrite the expression.

$5 \ln (r^{2}) + \frac{1}{2} (- \frac{50}{r}) + \frac{1}{2} (2 r) + C$

Step 18.4.2

Cancel the common factor of $2$ .

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Step 18.4.2.1

Move the leading negative in $- \frac{50}{r}$ into the numerator.

$5 \ln (r^{2}) + \frac{1}{2} \cdot \frac{- 50}{r} + \frac{1}{2} (2 r) + C$

Step 18.4.2.2

Factor $2$ out of $- 50$ .

$5 \ln (r^{2}) + \frac{1}{2} \cdot \frac{2 (- 25)}{r} + \frac{1}{2} (2 r) + C$

Step 18.4.2.3

Cancel the common factor.

$5 \ln (r^{2}) + \frac{1}{2} \cdot \frac{2 \cdot - 25}{r} + \frac{1}{2} (2 r) + C$

Step 18.4.2.4

Rewrite the expression.

$5 \ln (r^{2}) + \frac{- 25}{r} + \frac{1}{2} (2 r) + C$

Step 18.4.3

Cancel the common factor of $2$ .

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Step 18.4.3.1

Factor $2$ out of $2 r$ .

$5 \ln (r^{2}) + \frac{- 25}{r} + \frac{1}{2} (2 (r)) + C$

Step 18.4.3.2

Cancel the common factor.

$5 \ln (r^{2}) + \frac{- 25}{r} + \frac{1}{2} (2 r) + C$

Step 18.4.3.3

Rewrite the expression.

$5 \ln (r^{2}) + \frac{- 25}{r} + r + C$

Step 18.5

Move the negative in front of the fraction.

$5 \ln (r^{2}) - \frac{25}{r} + r + C$