Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Divide each term in $2 x \frac{d y}{d x} = 5 - 2 x^{3}$ by $2 x$ and simplify.

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Step 1.1.1

Divide each term in $2 x \frac{d y}{d x} = 5 - 2 x^{3}$ by $2 x$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

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.

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

Step 1.1.2.2.2

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

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

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 $- 2$ and $2$ .

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Step 1.1.3.1.1.1

Factor $2$ out of $- 2 x^{3}$ .

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

Step 1.1.3.1.1.2

Cancel the common factors.

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Step 1.1.3.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 1.1.3.1.1.2.2

Cancel the common factor.

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

Step 1.1.3.1.1.2.3

Rewrite the expression.

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

Step 1.1.3.1.2

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

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Step 1.1.3.1.2.1

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

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

Step 1.1.3.1.2.2

Cancel the common factors.

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Step 1.1.3.1.2.2.1

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

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

Step 1.1.3.1.2.2.2

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

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

Step 1.1.3.1.2.2.3

Cancel the common factor.

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

Step 1.1.3.1.2.2.4

Rewrite the expression.

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

Step 1.1.3.1.2.2.5

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

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

Step 1.2

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $\frac{5}{2}$ is constant with respect to $x$ , move $\frac{5}{2}$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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

Step 2.4

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

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