Calculus Examples

$3 x \frac{d y}{d x} = 5 - x^{3}$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $3 x \frac{d y}{d x} = 5 - x^{3}$ by $3 x$ and simplify.

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Step 1.1.1

Divide each term in $3 x \frac{d y}{d x} = 5 - x^{3}$ by $3 x$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $3$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

Step 1.1.2.2

Cancel the common factor of $x$ .

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Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

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

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

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

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Step 1.1.3.1.1.1

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

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

Step 1.1.3.1.1.2

Cancel the common factors.

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Step 1.1.3.1.1.2.1

Factor $x$ out of $3 x$ .

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

Step 1.1.3.1.1.2.2

Cancel the common factor.

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

Step 1.1.3.1.1.2.3

Rewrite the expression.

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

Step 1.1.3.1.2

Move the negative in front of the fraction.

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

Step 1.2

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $\frac{5}{3}$ is constant with respect to $x$ , move $\frac{5}{3}$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

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

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

Step 2.3.7.2.2

Multiply $3$ by $3$ .

$y + C_{1} = \frac{5}{3} \ln (| x |) - \frac{1}{9} x^{3} + C_{4}$

Step 2.4

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

$y = \frac{5}{3} \ln (| x |) - \frac{1}{9} x^{3} + C$