Calculus Examples

$\frac{d y}{d x} = \frac{(3 x^{3} + 2 x^{2} + 5) y}{x (4 y^{3} + 3 y)}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d x} = \frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{4 y^{3} + 3 y}$

Step 1.2

Multiply both sides by $\frac{4 y^{3} + 3 y}{y}$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = \frac{4 y^{3} + 3 y}{y} (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{4 y^{3} + 3 y})$

Step 1.3

Simplify.

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Step 1.3.1

Factor $y$ out of $4 y^{3} + 3 y$ .

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Step 1.3.1.1

Factor $y$ out of $4 y^{3}$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = \frac{y (4 y^{2}) + 3 y}{y} (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{4 y^{3} + 3 y})$

Step 1.3.1.2

Factor $y$ out of $3 y$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = \frac{y (4 y^{2}) + y \cdot 3}{y} (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{4 y^{3} + 3 y})$

Step 1.3.1.3

Factor $y$ out of $y (4 y^{2}) + y \cdot 3$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = \frac{y (4 y^{2} + 3)}{y} (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{4 y^{3} + 3 y})$

Step 1.3.2

Cancel the common factor of $y$ .

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Step 1.3.2.1

Cancel the common factor.

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = \frac{y (4 y^{2} + 3)}{y} (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{4 y^{3} + 3 y})$

Step 1.3.2.2

Divide $4 y^{2} + 3$ by $1$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = (4 y^{2} + 3) (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{4 y^{3} + 3 y})$

Step 1.3.3

Factor $y$ out of $4 y^{3} + 3 y$ .

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Step 1.3.3.1

Factor $y$ out of $4 y^{3}$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = (4 y^{2} + 3) (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{y (4 y^{2}) + 3 y})$

Step 1.3.3.2

Factor $y$ out of $3 y$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = (4 y^{2} + 3) (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{y (4 y^{2}) + y \cdot 3})$

Step 1.3.3.3

Factor $y$ out of $y (4 y^{2}) + y \cdot 3$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = (4 y^{2} + 3) (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{y (4 y^{2} + 3)})$

Step 1.3.4

Cancel the common factor of $y$ .

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Step 1.3.4.1

Cancel the common factor.

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = (4 y^{2} + 3) (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{y}{y (4 y^{2} + 3)})$

Step 1.3.4.2

Rewrite the expression.

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = (4 y^{2} + 3) (\frac{3 x^{3} + 2 x^{2} + 5}{x} \cdot \frac{1}{4 y^{2} + 3})$

Step 1.3.5

Multiply $\frac{3 x^{3} + 2 x^{2} + 5}{x}$ by $\frac{1}{4 y^{2} + 3}$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = (4 y^{2} + 3) \frac{3 x^{3} + 2 x^{2} + 5}{x (4 y^{2} + 3)}$

Step 1.3.6

Cancel the common factor of $4 y^{2} + 3$ .

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Step 1.3.6.1

Factor $4 y^{2} + 3$ out of $x (4 y^{2} + 3)$ .

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = (4 y^{2} + 3) \frac{3 x^{3} + 2 x^{2} + 5}{(4 y^{2} + 3) x}$

Step 1.3.6.2

Cancel the common factor.

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = (4 y^{2} + 3) \frac{3 x^{3} + 2 x^{2} + 5}{(4 y^{2} + 3) x}$

Step 1.3.6.3

Rewrite the expression.

$\frac{4 y^{3} + 3 y}{y} \frac{d y}{d x} = \frac{3 x^{3} + 2 x^{2} + 5}{x}$

Step 1.4

Rewrite the equation.

$\frac{4 y^{3} + 3 y}{y} d y = \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{4 y^{3} + 3 y}{y} d y = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the fraction into multiple fractions.

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Step 2.2.1.1

Factor $y$ out of $4 y^{3} + 3 y$ .

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Step 2.2.1.1.1

Factor $y$ out of $4 y^{3}$ .

$\int \frac{y (4 y^{2}) + 3 y}{y} d y = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.1.1.2

Factor $y$ out of $3 y$ .

$\int \frac{y (4 y^{2}) + y \cdot 3}{y} d y = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.1.1.3

Factor $y$ out of $y (4 y^{2}) + y \cdot 3$ .

$\int \frac{y (4 y^{2} + 3)}{y} d y = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.1.2

Cancel the common factor of $y$ .

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Step 2.2.1.2.1

Cancel the common factor.

$\int \frac{y (4 y^{2} + 3)}{y} d y = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.1.2.2

Divide $4 y^{2} + 3$ by $1$ .

$\int 4 y^{2} + 3 d y = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.2

Split the single integral into multiple integrals.

$\int 4 y^{2} d y + \int 3 d y = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.3

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

$4 \int y^{2} d y + \int 3 d y = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.4

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

$4 (\frac{1}{3} y^{3} + C_{1}) + \int 3 d y = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.5

Apply the constant rule.

$4 (\frac{1}{3} y^{3} + C_{1}) + 3 y + C_{2} = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.6

Simplify.

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Step 2.2.6.1

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

$4 (\frac{y^{3}}{3} + C_{1}) + 3 y + C_{2} = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.6.2

Simplify.

$\frac{4 y^{3}}{3} + 3 y + C_{3} = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.2.6.3

Reorder terms.

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int \frac{3 x^{3} + 2 x^{2} + 5}{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int \frac{3 x^{3}}{x} + \frac{2 x^{2}}{x} + \frac{5}{x} d x$

Step 2.3.2

Split the single integral into multiple integrals.

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int \frac{3 x^{3}}{x} d x + \int \frac{2 x^{2}}{x} d x + \int \frac{5}{x} d x$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Cancel the common factor of $x^{3}$ and $x$ .

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Step 2.3.3.1.1

Factor $x$ out of $3 x^{3}$ .

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int \frac{x (3 x^{2})}{x} d x + \int \frac{2 x^{2}}{x} d x + \int \frac{5}{x} d x$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

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

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int \frac{x (3 x^{2})}{x^{1}} d x + \int \frac{2 x^{2}}{x} d x + \int \frac{5}{x} d x$

Step 2.3.3.1.2.2

Factor $x$ out of $x^{1}$ .

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int \frac{x (3 x^{2})}{x \cdot 1} d x + \int \frac{2 x^{2}}{x} d x + \int \frac{5}{x} d x$

Step 2.3.3.1.2.3

Cancel the common factor.

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int \frac{x (3 x^{2})}{x \cdot 1} d x + \int \frac{2 x^{2}}{x} d x + \int \frac{5}{x} d x$

Step 2.3.3.1.2.4

Rewrite the expression.

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int \frac{3 x^{2}}{1} d x + \int \frac{2 x^{2}}{x} d x + \int \frac{5}{x} d x$

Step 2.3.3.1.2.5

Divide $3 x^{2}$ by $1$ .

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int 3 x^{2} d x + \int \frac{2 x^{2}}{x} d x + \int \frac{5}{x} d x$

Step 2.3.3.2

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

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Step 2.3.3.2.1

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

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int 3 x^{2} d x + \int \frac{x (2 x)}{x} d x + \int \frac{5}{x} d x$

Step 2.3.3.2.2

Cancel the common factors.

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Step 2.3.3.2.2.1

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

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int 3 x^{2} d x + \int \frac{x (2 x)}{x^{1}} d x + \int \frac{5}{x} d x$

Step 2.3.3.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int 3 x^{2} d x + \int \frac{x (2 x)}{x \cdot 1} d x + \int \frac{5}{x} d x$

Step 2.3.3.2.2.3

Cancel the common factor.

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int 3 x^{2} d x + \int \frac{x (2 x)}{x \cdot 1} d x + \int \frac{5}{x} d x$

Step 2.3.3.2.2.4

Rewrite the expression.

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int 3 x^{2} d x + \int \frac{2 x}{1} d x + \int \frac{5}{x} d x$

Step 2.3.3.2.2.5

Divide $2 x$ by $1$ .

$\frac{4}{3} y^{3} + 3 y + C_{3} = \int 3 x^{2} d x + \int 2 x d x + \int \frac{5}{x} d x$

Step 2.3.4

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

$\frac{4}{3} y^{3} + 3 y + C_{3} = 3 \int x^{2} d x + \int 2 x d x + \int \frac{5}{x} d x$

Step 2.3.5

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

$\frac{4}{3} y^{3} + 3 y + C_{3} = 3 (\frac{1}{3} x^{3} + C_{4}) + \int 2 x d x + \int \frac{5}{x} d x$

Step 2.3.6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{4}{3} y^{3} + 3 y + C_{3} = 3 (\frac{1}{3} x^{3} + C_{4}) + 2 \int x d x + \int \frac{5}{x} d x$

Step 2.3.7

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

$\frac{4}{3} y^{3} + 3 y + C_{3} = 3 (\frac{1}{3} x^{3} + C_{4}) + 2 (\frac{1}{2} x^{2} + C_{5}) + \int \frac{5}{x} d x$

Step 2.3.8

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\frac{4}{3} y^{3} + 3 y + C_{3} = 3 (\frac{1}{3} x^{3} + C_{4}) + 2 (\frac{1}{2} x^{2} + C_{5}) + 5 \int \frac{1}{x} d x$

Step 2.3.9

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

$\frac{4}{3} y^{3} + 3 y + C_{3} = 3 (\frac{1}{3} x^{3} + C_{4}) + 2 (\frac{1}{2} x^{2} + C_{5}) + 5 (\ln (| x |) + C_{6})$

Step 2.3.10

Simplify.

$\frac{4}{3} y^{3} + 3 y + C_{3} = x^{3} + x^{2} + 5 \ln (| x |) + C_{7}$

Step 2.4

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

$\frac{4}{3} y^{3} + 3 y = x^{3} + x^{2} + 5 \ln (| x |) + C$