Calculus Examples

$x + 3 y^{2} \sqrt{x^{2} + 1} \frac{d y}{d x} = 0$ , $y (0) = 1$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Subtract $x$ from both sides of the equation.

$3 y^{2} \sqrt{x^{2} + 1} \frac{d y}{d x} = - x$

Step 1.1.2

Divide each term in $3 y^{2} \sqrt{x^{2} + 1} \frac{d y}{d x} = - x$ by $3 y^{2} \sqrt{x^{2} + 1}$ and simplify.

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Step 1.1.2.1

Divide each term in $3 y^{2} \sqrt{x^{2} + 1} \frac{d y}{d x} = - x$ by $3 y^{2} \sqrt{x^{2} + 1}$ .

$\frac{3 y^{2} \sqrt{x^{2} + 1} \frac{d y}{d x}}{3 y^{2} \sqrt{x^{2} + 1}} = \frac{- x}{3 y^{2} \sqrt{x^{2} + 1}}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $3$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{3 y^{2} \sqrt{x^{2} + 1} \frac{d y}{d x}}{3 y^{2} \sqrt{x^{2} + 1}} = \frac{- x}{3 y^{2} \sqrt{x^{2} + 1}}$

Step 1.1.2.2.1.2

Rewrite the expression.

$\frac{y^{2} \sqrt{x^{2} + 1} \frac{d y}{d x}}{y^{2} \sqrt{x^{2} + 1}} = \frac{- x}{3 y^{2} \sqrt{x^{2} + 1}}$

Step 1.1.2.2.2

Cancel the common factor of $y^{2}$ .

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Step 1.1.2.2.2.1

Cancel the common factor.

$\frac{y^{2} \sqrt{x^{2} + 1} \frac{d y}{d x}}{y^{2} \sqrt{x^{2} + 1}} = \frac{- x}{3 y^{2} \sqrt{x^{2} + 1}}$

Step 1.1.2.2.2.2

Rewrite the expression.

$\frac{\sqrt{x^{2} + 1} \frac{d y}{d x}}{\sqrt{x^{2} + 1}} = \frac{- x}{3 y^{2} \sqrt{x^{2} + 1}}$

Step 1.1.2.2.3

Cancel the common factor of $\sqrt{x^{2} + 1}$ .

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Step 1.1.2.2.3.1

Cancel the common factor.

$\frac{\sqrt{x^{2} + 1} \frac{d y}{d x}}{\sqrt{x^{2} + 1}} = \frac{- x}{3 y^{2} \sqrt{x^{2} + 1}}$

Step 1.1.2.2.3.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{- x}{3 y^{2} \sqrt{x^{2} + 1}}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{x}{3 y^{2} \sqrt{x^{2} + 1}}$

Step 1.1.2.3.2

Multiply $\frac{x}{3 y^{2} \sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

$\frac{d y}{d x} = - (\frac{x}{3 y^{2} \sqrt{x^{2} + 1}} \cdot \frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}})$

Step 1.1.2.3.3

Combine and simplify the denominator.

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Step 1.1.2.3.3.1

Multiply $\frac{x}{3 y^{2} \sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} \sqrt{x^{2} + 1} \sqrt{x^{2} + 1}}$

Step 1.1.2.3.3.2

Move $\sqrt{x^{2} + 1}$ .

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} (\sqrt{x^{2} + 1} \sqrt{x^{2} + 1})}$

Step 1.1.2.3.3.3

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

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} ({\sqrt{x^{2} + 1}}^{1} \sqrt{x^{2} + 1})}$

Step 1.1.2.3.3.4

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

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} ({\sqrt{x^{2} + 1}}^{1} {\sqrt{x^{2} + 1}}^{1})}$

Step 1.1.2.3.3.5

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

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} {\sqrt{x^{2} + 1}}^{1 + 1}}$

Step 1.1.2.3.3.6

Add $1$ and $1$ .

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} {\sqrt{x^{2} + 1}}^{2}}$

Step 1.1.2.3.3.7

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

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Step 1.1.2.3.3.7.1

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

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} {({(x^{2} + 1)}^{\frac{1}{2}})}^{2}}$

Step 1.1.2.3.3.7.2

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

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} {(x^{2} + 1)}^{\frac{1}{2} \cdot 2}}$

Step 1.1.2.3.3.7.3

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

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} {(x^{2} + 1)}^{\frac{2}{2}}}$

Step 1.1.2.3.3.7.4

Cancel the common factor of $2$ .

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Step 1.1.2.3.3.7.4.1

Cancel the common factor.

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} {(x^{2} + 1)}^{\frac{2}{2}}}$

Step 1.1.2.3.3.7.4.2

Rewrite the expression.

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} {(x^{2} + 1)}^{1}}$

Step 1.1.2.3.3.7.5

Simplify.

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 y^{2} (x^{2} + 1)}$

Step 1.2

Regroup factors.

$\frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 (x^{2} + 1)} \cdot \frac{1}{y^{2}}$

Step 1.3

Multiply both sides by $y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} (- \frac{x \sqrt{x^{2} + 1}}{3 (x^{2} + 1)} \cdot \frac{1}{y^{2}})$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

$y^{2} \frac{d y}{d x} = - y^{2} (\frac{x \sqrt{x^{2} + 1}}{3 (x^{2} + 1)} \cdot \frac{1}{y^{2}})$

Step 1.4.2

Combine.

$y^{2} \frac{d y}{d x} = - y^{2} \frac{x \sqrt{x^{2} + 1} \cdot 1}{3 (x^{2} + 1) y^{2}}$

Step 1.4.3

Cancel the common factor of $y^{2}$ .

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Step 1.4.3.1

Factor $y^{2}$ out of $- y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} \cdot - 1 \frac{x \sqrt{x^{2} + 1} \cdot 1}{3 (x^{2} + 1) y^{2}}$

Step 1.4.3.2

Factor $y^{2}$ out of $3 (x^{2} + 1) y^{2}$ .

$y^{2} \frac{d y}{d x} = y^{2} \cdot - 1 \frac{x \sqrt{x^{2} + 1} \cdot 1}{y^{2} (3 (x^{2} + 1))}$

Step 1.4.3.3

Cancel the common factor.

$y^{2} \frac{d y}{d x} = y^{2} \cdot - 1 \frac{x \sqrt{x^{2} + 1} \cdot 1}{y^{2} (3 (x^{2} + 1))}$

Step 1.4.3.4

Rewrite the expression.

$y^{2} \frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1} \cdot 1}{3 (x^{2} + 1)}$

Step 1.4.4

Multiply $x$ by $1$ .

$y^{2} \frac{d y}{d x} = - \frac{x \sqrt{x^{2} + 1}}{3 (x^{2} + 1)}$

Step 1.5

Rewrite the equation.

$y^{2} d y = - \frac{x \sqrt{x^{2} + 1}}{3 (x^{2} + 1)} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

$\frac{1}{3} y^{3} + C_{1} = \int - \frac{x \sqrt{x^{2} + 1}}{3 (x^{2} + 1)} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\frac{1}{3} y^{3} + C_{1} = - \int \frac{x \sqrt{x^{2} + 1}}{3 (x^{2} + 1)} d x$

Step 2.3.2

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

$\frac{1}{3} y^{3} + C_{1} = - (\frac{1}{3} \int \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1} d x)$

Step 2.3.3

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

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Step 2.3.3.1

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

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Step 2.3.3.1.1

Differentiate $x^{2} + 1$ .

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

Step 2.3.3.1.2

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

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

Step 2.3.3.1.3

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

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

Step 2.3.3.1.4

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

$2 x + 0$

Step 2.3.3.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.3.2

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{3} \int \frac{\sqrt{u}}{u} \cdot \frac{1}{2} d u$

Step 2.3.4

Simplify.

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Step 2.3.4.1

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{3} \int \frac{\sqrt{u}}{u \cdot 2} d u$

Step 2.3.4.2

Move $2$ to the left of $u$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{3} \int \frac{\sqrt{u}}{2 u} d u$

Step 2.3.5

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

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

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Simplify.

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Step 2.3.6.1.1

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{2 \cdot 3} \int \frac{\sqrt{u}}{u} d u$

Step 2.3.6.1.2

Multiply $2$ by $3$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int \frac{\sqrt{u}}{u} d u$

Step 2.3.6.2

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int \frac{u^{\frac{1}{2}}}{u} d u$

Step 2.3.6.3

Simplify.

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Step 2.3.6.3.1

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int \frac{1}{u \cdot u^{- \frac{1}{2}}} d u$

Step 2.3.6.3.2

Multiply $u$ by $u^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.6.3.2.1

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

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Step 2.3.6.3.2.1.1

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int \frac{1}{u^{1} u^{- \frac{1}{2}}} d u$

Step 2.3.6.3.2.1.2

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int \frac{1}{u^{1 - \frac{1}{2}}} d u$

Step 2.3.6.3.2.2

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int \frac{1}{u^{\frac{2}{2} - \frac{1}{2}}} d u$

Step 2.3.6.3.2.3

Combine the numerators over the common denominator.

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int \frac{1}{u^{\frac{2 - 1}{2}}} d u$

Step 2.3.6.3.2.4

Subtract $1$ from $2$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 2.3.6.4

Apply basic rules of exponents.

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Step 2.3.6.4.1

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 2.3.6.4.2

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

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Step 2.3.6.4.2.1

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int u^{\frac{1}{2} \cdot - 1} d u$

Step 2.3.6.4.2.2

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int u^{\frac{- 1}{2}} d u$

Step 2.3.6.4.2.3

Move the negative in front of the fraction.

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \int u^{- \frac{1}{2}} d u$

Step 2.3.7

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} (2 u^{\frac{1}{2}} + C_{2})$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Rewrite $- \frac{1}{6} (2 u^{\frac{1}{2}} + C_{2})$ as $- \frac{1}{6} \cdot 2 u^{\frac{1}{2}} + C_{2}$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{6} \cdot 2 u^{\frac{1}{2}} + C_{2}$

Step 2.3.8.2

Simplify.

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Step 2.3.8.2.1

Multiply $2$ by $- 1$ .

$\frac{1}{3} y^{3} + C_{1} = - 2 (\frac{1}{6}) u^{\frac{1}{2}} + C_{2}$

Step 2.3.8.2.2

Combine $- 2$ and $\frac{1}{6}$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{- 2}{6} u^{\frac{1}{2}} + C_{2}$

Step 2.3.8.2.3

Cancel the common factor of $- 2$ and $6$ .

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Step 2.3.8.2.3.1

Factor $2$ out of $- 2$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{2 (- 1)}{6} u^{\frac{1}{2}} + C_{2}$

Step 2.3.8.2.3.2

Cancel the common factors.

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Step 2.3.8.2.3.2.1

Factor $2$ out of $6$ .

$\frac{1}{3} y^{3} + C_{1} = \frac{2 \cdot - 1}{2 \cdot 3} u^{\frac{1}{2}} + C_{2}$

Step 2.3.8.2.3.2.2

Cancel the common factor.

$\frac{1}{3} y^{3} + C_{1} = \frac{2 \cdot - 1}{2 \cdot 3} u^{\frac{1}{2}} + C_{2}$

Step 2.3.8.2.3.2.3

Rewrite the expression.

$\frac{1}{3} y^{3} + C_{1} = \frac{- 1}{3} u^{\frac{1}{2}} + C_{2}$

Step 2.3.8.2.4

Move the negative in front of the fraction.

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{3} u^{\frac{1}{2}} + C_{2}$

Step 2.3.9

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{3} {(x^{2} + 1)}^{\frac{1}{2}} + C_{2}$

Step 2.4

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

$\frac{1}{3} y^{3} = - \frac{1}{3} {(x^{2} + 1)}^{\frac{1}{2}} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $1$ for $y$ in $\frac{1}{3} y^{3} = - \frac{1}{3} {(x^{2} + 1)}^{\frac{1}{2}} + K$ .

$\frac{1}{3} \cdot 1^{3} = - \frac{1}{3} {(0^{2} + 1)}^{\frac{1}{2}} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \frac{1}{3} \cdot {(0^{2} + 1)}^{\frac{1}{2}} + K = \frac{1}{3} \cdot 1^{3}$ .

$- \frac{1}{3} \cdot {(0^{2} + 1)}^{\frac{1}{2}} + K = \frac{1}{3} \cdot 1^{3}$

Step 4.2

Simplify each term.

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Step 4.2.1

Raising $0$ to any positive power yields $0$ .

$- \frac{1}{3} \cdot {(0 + 1)}^{\frac{1}{2}} + K = \frac{1}{3} \cdot 1^{3}$

Step 4.2.2

Add $0$ and $1$ .

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

Step 4.2.3

One to any power is one.

$- \frac{1}{3} \cdot 1 + K = \frac{1}{3} \cdot 1^{3}$

Step 4.2.4

Multiply $- 1$ by $1$ .

$- \frac{1}{3} + K = \frac{1}{3} \cdot 1^{3}$

Step 4.3

Simplify $\frac{1}{3} \cdot 1^{3}$ .

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Step 4.3.1

One to any power is one.

$- \frac{1}{3} + K = \frac{1}{3} \cdot 1$

Step 4.3.2

Multiply $\frac{1}{3}$ by $1$ .

$- \frac{1}{3} + K = \frac{1}{3}$

Step 4.4

Move all terms not containing $K$ to the right side of the equation.

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Step 4.4.1

Add $\frac{1}{3}$ to both sides of the equation.

$K = \frac{1}{3} + \frac{1}{3}$

Step 4.4.2

Combine the numerators over the common denominator.

$K = \frac{1 + 1}{3}$

Step 4.4.3

Add $1$ and $1$ .

$K = \frac{2}{3}$

Step 5

Substitute $\frac{2}{3}$ for $K$ in $\frac{1}{3} y^{3} = - \frac{1}{3} {(x^{2} + 1)}^{\frac{1}{2}} + K$ and simplify.

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Step 5.1

Substitute $\frac{2}{3}$ for $K$ .

$\frac{1}{3} y^{3} = - \frac{1}{3} {(x^{2} + 1)}^{\frac{1}{2}} + \frac{2}{3}$

Step 5.2

Combine $\frac{1}{3}$ and $y^{3}$ .

$\frac{y^{3}}{3} = - \frac{1}{3} {(x^{2} + 1)}^{\frac{1}{2}} + \frac{2}{3}$

Step 5.3

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

$\frac{y^{3}}{3} = - \frac{{(x^{2} + 1)}^{\frac{1}{2}}}{3} + \frac{2}{3}$

Step 5.4

Combine the numerators over the common denominator.

$\frac{y^{3}}{3} = \frac{- {(x^{2} + 1)}^{\frac{1}{2}} + 2}{3}$