Calculus Examples

$\frac{d y}{d x} = \frac{9 + 20 x}{x y^{2}}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{9 + 20 x}{x} \cdot \frac{1}{y^{2}}$

Step 1.2

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

$y^{2} \frac{d y}{d x} = y^{2} (\frac{9 + 20 x}{x} \cdot \frac{1}{y^{2}})$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $\frac{9 + 20 x}{x}$ by $\frac{1}{y^{2}}$ .

$y^{2} \frac{d y}{d x} = y^{2} \frac{9 + 20 x}{x y^{2}}$

Step 1.3.2

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

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Step 1.3.2.1

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

$y^{2} \frac{d y}{d x} = y^{2} \frac{9 + 20 x}{y^{2} x}$

Step 1.3.2.2

Cancel the common factor.

$y^{2} \frac{d y}{d x} = y^{2} \frac{9 + 20 x}{y^{2} x}$

Step 1.3.2.3

Rewrite the expression.

$y^{2} \frac{d y}{d x} = \frac{9 + 20 x}{x}$

Step 1.4

Rewrite the equation.

$y^{2} d y = \frac{9 + 20 x}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} d y = \int \frac{9 + 20 x}{x} d x$

Step 2.2

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

$\frac{1}{3} y^{3} + C_{1} = \int \frac{9 + 20 x}{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

$\frac{1}{3} y^{3} + C_{1} = \int \frac{9}{x} + \frac{20 x}{x} d x$

Step 2.3.2

Split the single integral into multiple integrals.

$\frac{1}{3} y^{3} + C_{1} = \int \frac{9}{x} d x + \int \frac{20 x}{x} d x$

Step 2.3.3

Cancel the common factor of $x$ .

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Step 2.3.3.1

Cancel the common factor.

$\frac{1}{3} y^{3} + C_{1} = \int \frac{9}{x} d x + \int \frac{20 x}{x} d x$

Step 2.3.3.2

Divide $20$ by $1$ .

$\frac{1}{3} y^{3} + C_{1} = \int \frac{9}{x} d x + \int 20 d x$

Step 2.3.4

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$\frac{1}{3} y^{3} + C_{1} = 9 \int \frac{1}{x} d x + \int 20 d x$

Step 2.3.5

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

$\frac{1}{3} y^{3} + C_{1} = 9 (\ln (| x |) + C_{2}) + \int 20 d x$

Step 2.3.6

Apply the constant rule.

$\frac{1}{3} y^{3} + C_{1} = 9 (\ln (| x |) + C_{2}) + 20 x + C_{3}$

Step 2.3.7

Simplify.

$\frac{1}{3} y^{3} + C_{1} = 9 \ln (| x |) + 20 x + C_{4}$

Step 2.3.8

Reorder terms.

$\frac{1}{3} y^{3} + C_{1} = 20 x + 9 \ln (| x |) + C_{4}$

Step 2.4

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

$\frac{1}{3} y^{3} = 20 x + 9 \ln (| x |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (20 x + 9 \ln (| x |) + C)$

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 $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (20 x + 9 \ln (| x |) + C)$

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.

$3 \frac{y^{3}}{3} = 3 (20 x + 9 \ln (| x |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (20 x + 9 \ln (| x |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (20 x + 9 \ln (| x |) + C)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$y^{3} = 3 (20 x) + 3 (9 \ln (| x |)) + 3 C$

Step 3.2.2.1.2

Simplify.

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Step 3.2.2.1.2.1

Multiply $20$ by $3$ .

$y^{3} = 60 x + 3 (9 \ln (| x |)) + 3 C$

Step 3.2.2.1.2.2

Multiply $9$ by $3$ .

$y^{3} = 60 x + 27 \ln (| x |) + 3 C$

Step 3.3

Simplify $27 \ln (| x |)$ by moving $27$ inside the logarithm.

$y^{3} = 60 x + \ln ({| x |}^{27}) + 3 C$

Step 3.4

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

$y = \sqrt[3]{60 x + \ln ({| x |}^{27}) + 3 C}$

Step 4

Simplify the constant of integration.

$y = \sqrt[3]{60 x + \ln ({| x |}^{27}) + C}$