Calculus Examples

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

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

Reorder factors in $x^{3} + {(y + 1)}^{2} \frac{d y}{d x}$ .

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

Step 1.1.2

Subtract $x^{3}$ from both sides of the equation.

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of ${(y + 1)}^{2}$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

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

Step 1.1.3.2.1.2

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

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

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Move the negative in front of the fraction.

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

Step 1.2

Multiply both sides by ${(y + 1)}^{2}$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.2

Cancel the common factor of ${(y + 1)}^{2}$ .

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Step 1.3.2.1

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

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = y + 1$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = y + 1$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $y + 1$ .

$\frac{d}{d y} [y + 1]$

Step 2.2.1.1.2

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

$\frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 2.2.1.1.3

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

$1 + \frac{d}{d y} [1]$

Step 2.2.1.1.4

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

$1 + 0$

Step 2.2.1.1.5

Add $1$ and $0$ .

$1$

Step 2.2.1.2

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

$\int u^{2} d u = \int - x^{3} d x$

Step 2.2.2

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

$\frac{1}{3} u^{3} + C_{1} = \int - x^{3} d x$

Step 2.2.3

Replace all occurrences of $u$ with $y + 1$ .

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

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.

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

Step 2.3.2

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 2.3.3

Rewrite $- (\frac{1}{4} x^{4} + C_{2})$ as $- \frac{1}{4} x^{4} + C_{2}$ .

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

Step 2.4

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

$\frac{1}{3} {(y + 1)}^{3} = - \frac{1}{4} x^{4} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} {(y + 1)}^{3}) = 3 (- \frac{1}{4} x^{4} + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $3 (\frac{1}{3} {(y + 1)}^{3})$ .

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Step 3.2.1.1.1

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

$3 \frac{{(y + 1)}^{3}}{3} = 3 (- \frac{1}{4} x^{4} + K)$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$3 \frac{{(y + 1)}^{3}}{3} = 3 (- \frac{1}{4} x^{4} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

${(y + 1)}^{3} = 3 (- \frac{1}{4} x^{4} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (- \frac{1}{4} x^{4} + K)$ .

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Step 3.2.2.1.1

Combine $x^{4}$ and $\frac{1}{4}$ .

${(y + 1)}^{3} = 3 (- \frac{x^{4}}{4} + K)$

Step 3.2.2.1.2

Apply the distributive property.

${(y + 1)}^{3} = 3 (- \frac{x^{4}}{4}) + 3 K$

Step 3.2.2.1.3

Multiply $3 (- \frac{x^{4}}{4})$ .

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Step 3.2.2.1.3.1

Multiply $- 1$ by $3$ .

${(y + 1)}^{3} = - 3 \frac{x^{4}}{4} + 3 K$

Step 3.2.2.1.3.2

Combine $- 3$ and $\frac{x^{4}}{4}$ .

${(y + 1)}^{3} = \frac{- 3 x^{4}}{4} + 3 K$

Step 3.2.2.1.4

Move the negative in front of the fraction.

${(y + 1)}^{3} = - \frac{3 x^{4}}{4} + 3 K$

Step 3.3

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

$y + 1 = \sqrt[3]{- \frac{3 x^{4}}{4} + 3 K}$

Step 3.4

Simplify $\sqrt[3]{- \frac{3 x^{4}}{4} + 3 K}$ .

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Step 3.4.1

Rewrite.

$y + 1 = 0 + 0 + \sqrt[3]{- \frac{3 x^{4}}{4} + 3 K}$

Step 3.4.2

Simplify by adding zeros.

$y + 1 = \sqrt[3]{- \frac{3 x^{4}}{4} + 3 K}$

Step 3.4.3

Factor $3$ out of $- \frac{3 x^{4}}{4} + 3 K$ .

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Step 3.4.3.1

Factor $3$ out of $- \frac{3 x^{4}}{4}$ .

$y + 1 = \sqrt[3]{3 (- \frac{x^{4}}{4}) + 3 K}$

Step 3.4.3.2

Factor $3$ out of $3 K$ .

$y + 1 = \sqrt[3]{3 (- \frac{x^{4}}{4}) + 3 K}$

Step 3.4.3.3

Factor $3$ out of $3 (- \frac{x^{4}}{4}) + 3 K$ .

$y + 1 = \sqrt[3]{3 (- \frac{x^{4}}{4} + K)}$

Step 3.4.4

To write $K$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y + 1 = \sqrt[3]{3 (- \frac{x^{4}}{4} + K \cdot \frac{4}{4})}$

Step 3.4.5

Simplify terms.

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Step 3.4.5.1

Combine $K$ and $\frac{4}{4}$ .

$y + 1 = \sqrt[3]{3 (- \frac{x^{4}}{4} + \frac{K \cdot 4}{4})}$

Step 3.4.5.2

Combine the numerators over the common denominator.

$y + 1 = \sqrt[3]{3 \frac{- x^{4} + K \cdot 4}{4}}$

Step 3.4.6

Move $4$ to the left of $K$ .

$y + 1 = \sqrt[3]{3 \frac{- x^{4} + 4 K}{4}}$

Step 3.4.7

Combine $3$ and $\frac{- x^{4} + 4 K}{4}$ .

$y + 1 = \sqrt[3]{\frac{3 (- x^{4} + 4 K)}{4}}$

Step 3.4.8

Rewrite $\sqrt[3]{\frac{3 (- x^{4} + 4 K)}{4}}$ as $\frac{\sqrt[3]{3 (- x^{4} + 4 K)}}{\sqrt[3]{4}}$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)}}{\sqrt[3]{4}}$

Step 3.4.9

Multiply $\frac{\sqrt[3]{3 (- x^{4} + 4 K)}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)}}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$

Step 3.4.10

Combine and simplify the denominator.

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Step 3.4.10.1

Multiply $\frac{\sqrt[3]{3 (- x^{4} + 4 K)}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{\sqrt[3]{4} {\sqrt[3]{4}}^{2}}$

Step 3.4.10.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 3.4.10.3

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

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 3.4.10.4

Add $1$ and $2$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{3}}$

Step 3.4.10.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 3.4.10.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 3.4.10.5.2

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

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 3.4.10.5.3

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

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.4.10.5.4

Cancel the common factor of $3$ .

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Step 3.4.10.5.4.1

Cancel the common factor.

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{\frac{3}{3}}}$

Step 3.4.10.5.4.2

Rewrite the expression.

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 3.4.10.5.5

Evaluate the exponent.

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} {\sqrt[3]{4}}^{2}}{4}$

Step 3.4.11

Simplify the numerator.

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Step 3.4.11.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} \sqrt[3]{4^{2}}}{4}$

Step 3.4.11.2

Raise $4$ to the power of $2$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} \sqrt[3]{16}}{4}$

Step 3.4.11.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 3.4.11.3.1

Factor $8$ out of $16$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} \sqrt[3]{8 (2)}}{4}$

Step 3.4.11.3.2

Rewrite $8$ as $2^{3}$ .

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} \sqrt[3]{2^{3} \cdot 2}}{4}$

Step 3.4.11.4

Pull terms out from under the radical.

$y + 1 = \frac{\sqrt[3]{3 (- x^{4} + 4 K)} \cdot 2 \sqrt[3]{2}}{4}$

Step 3.4.11.5

Combine exponents.

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Step 3.4.11.5.1

Combine using the product rule for radicals.

$y + 1 = \frac{2 \sqrt[3]{3 (- x^{4} + 4 K) \cdot 2}}{4}$

Step 3.4.11.5.2

Multiply $2$ by $3$ .

$y + 1 = \frac{2 \sqrt[3]{6 (- x^{4} + 4 K)}}{4}$

Step 3.4.12

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

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Step 3.4.12.1

Factor $2$ out of $2 \sqrt[3]{6 (- x^{4} + 4 K)}$ .

$y + 1 = \frac{2 \sqrt[3]{6 (- x^{4} + 4 K)}}{4}$

Step 3.4.12.2

Cancel the common factors.

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Step 3.4.12.2.1

Factor $2$ out of $4$ .

$y + 1 = \frac{2 \sqrt[3]{6 (- x^{4} + 4 K)}}{2 \cdot 2}$

Step 3.4.12.2.2

Cancel the common factor.

$y + 1 = \frac{2 \sqrt[3]{6 (- x^{4} + 4 K)}}{2 \cdot 2}$

Step 3.4.12.2.3

Rewrite the expression.

$y + 1 = \frac{\sqrt[3]{6 (- x^{4} + 4 K)}}{2}$

Step 3.5

Subtract $1$ from both sides of the equation.

$y = \frac{\sqrt[3]{6 (- x^{4} + 4 K)}}{2} - 1$

Step 3.6

Simplify $\frac{\sqrt[3]{6 (- x^{4} + 4 K)}}{2} - 1$ .

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Step 3.6.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{\sqrt[3]{6 (- x^{4} + 4 K)}}{2} - 1 \cdot \frac{2}{2}$

Step 3.6.2

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

$y = \frac{\sqrt[3]{6 (- x^{4} + 4 K)}}{2} + \frac{- 1 \cdot 2}{2}$

Step 3.6.3

Combine the numerators over the common denominator.

$y = \frac{\sqrt[3]{6 (- x^{4} + 4 K)} - 1 \cdot 2}{2}$

Step 3.6.4

Multiply $- 1$ by $2$ .

$y = \frac{\sqrt[3]{6 (- x^{4} + 4 K)} - 2}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{6 (- x^{4} + K)} - 2}{2}$