Calculus Examples

$\frac{d u}{d v} = \frac{3 v \sqrt{1 + u^{2}}}{u}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d u}{d v} = 3 v \frac{\sqrt{1 + u^{2}}}{u}$

Step 1.2

Multiply both sides by $\frac{u}{\sqrt{1 + u^{2}}}$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{u}{\sqrt{1 + u^{2}}} (3 v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u}{\sqrt{1 + u^{2}}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.2

Multiply $\frac{u}{\sqrt{1 + u^{2}}}$ by $\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 (\frac{u}{\sqrt{1 + u^{2}}} \cdot \frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}) (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3

Combine and simplify the denominator.

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Step 1.3.3.1

Multiply $\frac{u}{\sqrt{1 + u^{2}}}$ by $\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}} \sqrt{1 + u^{2}}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.2

Raise $\sqrt{1 + u^{2}}$ to the power of $1$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} \sqrt{1 + u^{2}}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.3

Raise $\sqrt{1 + u^{2}}$ to the power of $1$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} {\sqrt{1 + u^{2}}}^{1}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.4

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

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1 + 1}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.5

Add $1$ and $1$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{2}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.6

Rewrite ${\sqrt{1 + u^{2}}}^{2}$ as $1 + u^{2}$ .

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Step 1.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + u^{2}}$ as ${(1 + u^{2})}^{\frac{1}{2}}$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{{({(1 + u^{2})}^{\frac{1}{2}})}^{2}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.6.2

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

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{1}{2} \cdot 2}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.6.3

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

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{2}{2}}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.3.6.4.1

Cancel the common factor.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{2}{2}}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.6.4.2

Rewrite the expression.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{1}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.3.6.5

Simplify.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 \frac{u \sqrt{1 + u^{2}}}{1 + u^{2}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.4

Combine $3$ and $\frac{u \sqrt{1 + u^{2}}}{1 + u^{2}}$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{3 (u \sqrt{1 + u^{2}})}{1 + u^{2}} (v \frac{\sqrt{1 + u^{2}}}{u})$

Step 1.3.5

Combine $v$ and $\frac{\sqrt{1 + u^{2}}}{u}$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{3 u \sqrt{1 + u^{2}}}{1 + u^{2}} \cdot \frac{v \sqrt{1 + u^{2}}}{u}$

Step 1.3.6

Cancel the common factor of $u$ .

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Step 1.3.6.1

Factor $u$ out of $3 u \sqrt{1 + u^{2}}$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{u (3 \sqrt{1 + u^{2}})}{1 + u^{2}} \cdot \frac{v \sqrt{1 + u^{2}}}{u}$

Step 1.3.6.2

Cancel the common factor.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{u (3 \sqrt{1 + u^{2}})}{1 + u^{2}} \cdot \frac{v \sqrt{1 + u^{2}}}{u}$

Step 1.3.6.3

Rewrite the expression.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{3 \sqrt{1 + u^{2}}}{1 + u^{2}} (v \sqrt{1 + u^{2}})$

Step 1.3.7

Combine $v$ and $\frac{3 \sqrt{1 + u^{2}}}{1 + u^{2}}$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v (3 \sqrt{1 + u^{2}})}{1 + u^{2}} \sqrt{1 + u^{2}}$

Step 1.3.8

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

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v (3 \sqrt{1 + u^{2}}) \sqrt{1 + u^{2}}}{1 + u^{2}}$

Step 1.3.9

Raise $\sqrt{1 + u^{2}}$ to the power of $1$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v (3 ({\sqrt{1 + u^{2}}}^{1} \sqrt{1 + u^{2}}))}{1 + u^{2}}$

Step 1.3.10

Raise $\sqrt{1 + u^{2}}$ to the power of $1$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v (3 ({\sqrt{1 + u^{2}}}^{1} {\sqrt{1 + u^{2}}}^{1}))}{1 + u^{2}}$

Step 1.3.11

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

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v (3 {\sqrt{1 + u^{2}}}^{1 + 1})}{1 + u^{2}}$

Step 1.3.12

Add $1$ and $1$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v (3 {\sqrt{1 + u^{2}}}^{2})}{1 + u^{2}}$

Step 1.3.13

Rewrite ${\sqrt{1 + u^{2}}}^{2}$ as $1 + u^{2}$ .

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Step 1.3.13.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + u^{2}}$ as ${(1 + u^{2})}^{\frac{1}{2}}$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v \cdot 3 {({(1 + u^{2})}^{\frac{1}{2}})}^{2}}{1 + u^{2}}$

Step 1.3.13.2

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

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v \cdot 3 {(1 + u^{2})}^{\frac{1}{2} \cdot 2}}{1 + u^{2}}$

Step 1.3.13.3

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

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v \cdot 3 {(1 + u^{2})}^{\frac{2}{2}}}{1 + u^{2}}$

Step 1.3.13.4

Cancel the common factor of $2$ .

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Step 1.3.13.4.1

Cancel the common factor.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v \cdot 3 {(1 + u^{2})}^{\frac{2}{2}}}{1 + u^{2}}$

Step 1.3.13.4.2

Rewrite the expression.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v \cdot 3 {(1 + u^{2})}^{1}}{1 + u^{2}}$

Step 1.3.13.5

Simplify.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v \cdot 3 (1 + u^{2})}{1 + u^{2}}$

Step 1.3.14

Cancel the common factor of $1 + u^{2}$ .

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Step 1.3.14.1

Cancel the common factor.

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = \frac{v \cdot 3 (1 + u^{2})}{1 + u^{2}}$

Step 1.3.14.2

Divide $v \cdot 3$ by $1$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = v \cdot 3$

Step 1.3.15

Move $3$ to the left of $v$ .

$\frac{u}{\sqrt{1 + u^{2}}} \frac{d u}{d v} = 3 v$

Step 1.4

Rewrite the equation.

$\frac{u}{\sqrt{1 + u^{2}}} d u = 3 v d v$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{u}{\sqrt{1 + u^{2}}} d u = \int 3 v d v$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Differentiate $1 + u^{2}$ .

$\frac{d}{d u} [1 + u^{2}]$

Step 2.2.1.1.2

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

$\frac{d}{d u} [1] + \frac{d}{d u} [u^{2}]$

Step 2.2.1.1.3

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

$0 + \frac{d}{d u} [u^{2}]$

Step 2.2.1.1.4

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

$0 + 2 u$

Step 2.2.1.1.5

Add $0$ and $2 u$ .

$2 u$

Step 2.2.1.2

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

$\int \frac{1}{\sqrt{u_{1}}} \cdot \frac{1}{2} d u_{1} = \int 3 v d v$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Multiply $\frac{1}{\sqrt{u_{1}}}$ by $\frac{1}{2}$ .

$\int \frac{1}{\sqrt{u_{1}} \cdot 2} d u_{1} = \int 3 v d v$

Step 2.2.2.2

Move $2$ to the left of $\sqrt{u_{1}}$ .

$\int \frac{1}{2 \sqrt{u_{1}}} d u_{1} = \int 3 v d v$

Step 2.2.3

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

$\frac{1}{2} \int \frac{1}{\sqrt{u_{1}}} d u_{1} = \int 3 v d v$

Step 2.2.4

Apply basic rules of exponents.

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Step 2.2.4.1

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

$\frac{1}{2} \int \frac{1}{{u_{1}}^{\frac{1}{2}}} d u_{1} = \int 3 v d v$

Step 2.2.4.2

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

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

Step 2.2.4.3

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

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Step 2.2.4.3.1

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

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

Step 2.2.4.3.2

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

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

Step 2.2.4.3.3

Move the negative in front of the fraction.

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

Step 2.2.5

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

$\frac{1}{2} (2 {u_{1}}^{\frac{1}{2}} + C_{1}) = \int 3 v d v$

Step 2.2.6

Simplify.

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Step 2.2.6.1

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

$\frac{1}{2} \cdot 2 {u_{1}}^{\frac{1}{2}} + C_{1} = \int 3 v d v$

Step 2.2.6.2

Simplify.

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Step 2.2.6.2.1

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

$\frac{2}{2} {u_{1}}^{\frac{1}{2}} + C_{1} = \int 3 v d v$

Step 2.2.6.2.2

Cancel the common factor of $2$ .

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Step 2.2.6.2.2.1

Cancel the common factor.

$\frac{2}{2} {u_{1}}^{\frac{1}{2}} + C_{1} = \int 3 v d v$

Step 2.2.6.2.2.2

Rewrite the expression.

$1 {u_{1}}^{\frac{1}{2}} + C_{1} = \int 3 v d v$

Step 2.2.6.2.3

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

${u_{1}}^{\frac{1}{2}} + C_{1} = \int 3 v d v$

Step 2.2.7

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

${(1 + u^{2})}^{\frac{1}{2}} + C_{1} = \int 3 v d v$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $3$ is constant with respect to $v$ , move $3$ out of the integral.

${(1 + u^{2})}^{\frac{1}{2}} + C_{1} = 3 \int v d v$

Step 2.3.2

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

${(1 + u^{2})}^{\frac{1}{2}} + C_{1} = 3 (\frac{1}{2} v^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

${(1 + u^{2})}^{\frac{1}{2}} + C_{1} = 3 (\frac{1}{2}) v^{2} + C_{2}$

Step 2.3.3.2

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

${(1 + u^{2})}^{\frac{1}{2}} + C_{1} = \frac{3}{2} v^{2} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

${(1 + u^{2})}^{\frac{1}{2}} = \frac{3}{2} v^{2} + K$

Step 3

Solve for $u$ .

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Step 3.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(1 + u^{2})}^{\frac{1}{2}})}^{2} = {(\frac{3}{2} v^{2} + K)}^{2}$

Step 3.2

Simplify the exponent.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify ${({(1 + u^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

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Step 3.2.1.1.1.1

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

${(1 + u^{2})}^{\frac{1}{2} \cdot 2} = {(\frac{3}{2} v^{2} + K)}^{2}$

Step 3.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.2.1

Cancel the common factor.

${(1 + u^{2})}^{\frac{1}{2} \cdot 2} = {(\frac{3}{2} v^{2} + K)}^{2}$

Step 3.2.1.1.1.2.2

Rewrite the expression.

${(1 + u^{2})}^{1} = {(\frac{3}{2} v^{2} + K)}^{2}$

Step 3.2.1.1.2

Simplify.

$1 + u^{2} = {(\frac{3}{2} v^{2} + K)}^{2}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify ${(\frac{3}{2} v^{2} + K)}^{2}$ .

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Step 3.2.2.1.1

Combine fractions.

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Step 3.2.2.1.1.1

Combine $\frac{3}{2}$ and $v^{2}$ .

$1 + u^{2} = {(\frac{3 v^{2}}{2} + K)}^{2}$

Step 3.2.2.1.1.2

Rewrite ${(\frac{3 v^{2}}{2} + K)}^{2}$ as $(\frac{3 v^{2}}{2} + K) (\frac{3 v^{2}}{2} + K)$ .

$1 + u^{2} = (\frac{3 v^{2}}{2} + K) (\frac{3 v^{2}}{2} + K)$

Step 3.2.2.1.2

Expand $(\frac{3 v^{2}}{2} + K) (\frac{3 v^{2}}{2} + K)$ using the FOIL Method.

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Step 3.2.2.1.2.1

Apply the distributive property.

$1 + u^{2} = \frac{3 v^{2}}{2} (\frac{3 v^{2}}{2} + K) + K (\frac{3 v^{2}}{2} + K)$

Step 3.2.2.1.2.2

Apply the distributive property.

$1 + u^{2} = \frac{3 v^{2}}{2} \cdot \frac{3 v^{2}}{2} + \frac{3 v^{2}}{2} K + K (\frac{3 v^{2}}{2} + K)$

Step 3.2.2.1.2.3

Apply the distributive property.

$1 + u^{2} = \frac{3 v^{2}}{2} \cdot \frac{3 v^{2}}{2} + \frac{3 v^{2}}{2} K + K \frac{3 v^{2}}{2} + K \cdot K$

Step 3.2.2.1.3

Simplify and combine like terms.

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Step 3.2.2.1.3.1

Simplify each term.

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Step 3.2.2.1.3.1.1

Combine.

$1 + u^{2} = \frac{3 v^{2} (3 v^{2})}{2 \cdot 2} + \frac{3 v^{2}}{2} K + K \frac{3 v^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.2

Multiply $v^{2}$ by $v^{2}$ by adding the exponents.

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Step 3.2.2.1.3.1.2.1

Move $v^{2}$ .

$1 + u^{2} = \frac{3 (v^{2} v^{2}) \cdot 3}{2 \cdot 2} + \frac{3 v^{2}}{2} K + K \frac{3 v^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.2.2

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

$1 + u^{2} = \frac{3 v^{2 + 2} \cdot 3}{2 \cdot 2} + \frac{3 v^{2}}{2} K + K \frac{3 v^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.2.3

Add $2$ and $2$ .

$1 + u^{2} = \frac{3 v^{4} \cdot 3}{2 \cdot 2} + \frac{3 v^{2}}{2} K + K \frac{3 v^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.3

Multiply $3$ by $3$ .

$1 + u^{2} = \frac{9 v^{4}}{2 \cdot 2} + \frac{3 v^{2}}{2} K + K \frac{3 v^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.4

Multiply $2$ by $2$ .

$1 + u^{2} = \frac{9 v^{4}}{4} + \frac{3 v^{2}}{2} K + K \frac{3 v^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.5

Combine $\frac{3 v^{2}}{2}$ and $K$ .

$1 + u^{2} = \frac{9 v^{4}}{4} + \frac{3 v^{2} K}{2} + K \frac{3 v^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.6

Combine $K$ and $\frac{3 v^{2}}{2}$ .

$1 + u^{2} = \frac{9 v^{4}}{4} + \frac{3 v^{2} K}{2} + \frac{K (3 v^{2})}{2} + K (K)$

Step 3.2.2.1.3.1.7

Move $3$ to the left of $K$ .

$1 + u^{2} = \frac{9 v^{4}}{4} + \frac{3 v^{2} K}{2} + \frac{3 \cdot K v^{2}}{2} + K \cdot K$

Step 3.2.2.1.3.1.8

Multiply $K$ by $K$ .

$1 + u^{2} = \frac{9 v^{4}}{4} + \frac{3 v^{2} K}{2} + \frac{3 K v^{2}}{2} + K^{2}$

Step 3.2.2.1.3.2

Add $\frac{3 v^{2} K}{2}$ and $\frac{3 K v^{2}}{2}$ .

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Step 3.2.2.1.3.2.1

Move $v^{2}$ .

$1 + u^{2} = \frac{9 v^{4}}{4} + \frac{3 K v^{2}}{2} + \frac{3 K v^{2}}{2} + K^{2}$

Step 3.2.2.1.3.2.2

Add $\frac{3 K v^{2}}{2}$ and $\frac{3 K v^{2}}{2}$ .

$1 + u^{2} = \frac{9 v^{4}}{4} + 2 \frac{3 K v^{2}}{2} + K^{2}$

Step 3.2.2.1.4

Cancel the common factor of $2$ .

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Step 3.2.2.1.4.1

Cancel the common factor.

$1 + u^{2} = \frac{9 v^{4}}{4} + 2 \frac{3 K v^{2}}{2} + K^{2}$

Step 3.2.2.1.4.2

Rewrite the expression.

$1 + u^{2} = \frac{9 v^{4}}{4} + 3 K v^{2} + K^{2}$

Step 3.3

Solve for $u$ .

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Step 3.3.1

Subtract $1$ from both sides of the equation.

$u^{2} = \frac{9 v^{4}}{4} + 3 K v^{2} + K^{2} - 1$

Step 3.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$u = \pm \sqrt{\frac{9 v^{4}}{4} + 3 K v^{2} + K^{2} - 1}$

Step 3.3.3

Simplify $\pm \sqrt{\frac{9 v^{4}}{4} + 3 K v^{2} + K^{2} - 1}$ .

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Step 3.3.3.1

Factor using the perfect square rule.

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Step 3.3.3.1.1

Rewrite $9 v^{4}$ as ${(3 v^{2})}^{2}$ .

$u = \pm \sqrt{\frac{{(3 v^{2})}^{2}}{4} + 3 K v^{2} + K^{2} - 1}$

Step 3.3.3.1.2

Rewrite $4$ as $2^{2}$ .

$u = \pm \sqrt{\frac{{(3 v^{2})}^{2}}{2^{2}} + 3 K v^{2} + K^{2} - 1}$

Step 3.3.3.1.3

Rewrite $\frac{{(3 v^{2})}^{2}}{2^{2}}$ as ${(\frac{3 v^{2}}{2})}^{2}$ .

$u = \pm \sqrt{{(\frac{3 v^{2}}{2})}^{2} + 3 K v^{2} + K^{2} - 1}$

Step 3.3.3.1.4

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$3 K v^{2} = 2 \cdot \frac{3 v^{2}}{2} \cdot K$

Step 3.3.3.1.5

Rewrite the polynomial.

$u = \pm \sqrt{{(\frac{3 v^{2}}{2})}^{2} + 2 \cdot \frac{3 v^{2}}{2} \cdot K + K^{2} - 1}$

Step 3.3.3.1.6

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \frac{3 v^{2}}{2}$ and $b = K$ .

$u = \pm \sqrt{{(\frac{3 v^{2}}{2} + K)}^{2} - 1}$

Step 3.3.3.2

Rewrite $1$ as $1^{2}$ .

$u = \pm \sqrt{{(\frac{3 v^{2}}{2} + K)}^{2} - 1^{2}}$

Step 3.3.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{3 v^{2}}{2} + K$ and $b = 1$ .

$u = \pm \sqrt{(\frac{3 v^{2}}{2} + K + 1) (\frac{3 v^{2}}{2} + K - 1)}$

Step 3.3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.4.1

First, use the positive value of the $\pm$ to find the first solution.