Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

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

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Factor $y^{2}$ out of $y^{2} + x y^{2}$ .

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Step 1.1.3.1.1

Multiply by $1$ .

$\frac{d y}{d x} = \frac{1}{y^{2} \cdot 1 + x y^{2}}$

Step 1.1.3.1.2

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

$\frac{d y}{d x} = \frac{1}{y^{2} \cdot 1 + y^{2} x}$

Step 1.1.3.1.3

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} x$ .

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

Step 1.2

Regroup factors.

$\frac{d y}{d x} = \frac{1}{y^{2}} \cdot \frac{1}{1 + x}$

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

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

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Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} d y = \int \frac{1}{1 + x} d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Differentiate $1 + x$ .

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

Step 2.3.1.1.2

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 2.3.1.1.3

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 2.3.1.1.4

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

$0 + 1$

Step 2.3.1.1.5

Add $0$ and $1$ .

$1$

Step 2.3.1.2

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

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

Step 2.3.2

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

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

Step 2.3.3

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

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

Step 2.4

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

$\frac{1}{3} y^{3} = \ln (| 1 + x |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (\ln (| 1 + x |) + C)$

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

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Step 3.2.1.1.1

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

$3 \frac{y^{3}}{3} = 3 (\ln (| 1 + x |) + C)$

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.

$3 \frac{y^{3}}{3} = 3 (\ln (| 1 + x |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (\ln (| 1 + x |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$y^{3} = 3 \ln (| 1 + x |) + 3 C$

Step 3.3

Simplify $3 \ln (| 1 + x |)$ by moving $3$ inside the logarithm.

$y^{3} = \ln ({| 1 + x |}^{3}) + 3 C$

Step 3.4

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

$y = \sqrt[3]{\ln ({| 1 + x |}^{3}) + 3 C}$

Step 4

Simplify the constant of integration.

$y = \sqrt[3]{\ln ({| 1 + x |}^{3}) + C}$