Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Factor $y^{2}$ out of $3 y^{2}$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

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

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

Step 2.3.4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.3.7

Apply the constant rule.

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

Step 2.3.8

Simplify.

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Step 2.3.8.1

Simplify.

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Step 2.3.8.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

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

Step 2.3.8.1.2

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

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

Step 2.3.8.2

Simplify.

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

Step 2.4

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

$\frac{1}{3} y^{3} = \frac{1}{3} (x^{4} + x^{3} + x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (\frac{1}{3} (x^{4} + x^{3} + x) + K)$

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 $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (\frac{1}{3} (x^{4} + x^{3} + x) + K)$

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.

$3 \frac{y^{3}}{3} = 3 (\frac{1}{3} (x^{4} + x^{3} + x) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (\frac{1}{3} (x^{4} + x^{3} + x) + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (\frac{1}{3} (x^{4} + x^{3} + x) + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$y^{3} = 3 (\frac{1}{3} x^{4} + \frac{1}{3} x^{3} + \frac{1}{3} x + K)$

Step 3.2.2.1.1.2

Simplify.

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Step 3.2.2.1.1.2.1

Combine $\frac{1}{3}$ and $x^{4}$ .

$y^{3} = 3 (\frac{x^{4}}{3} + \frac{1}{3} x^{3} + \frac{1}{3} x + K)$

Step 3.2.2.1.1.2.2

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

$y^{3} = 3 (\frac{x^{4}}{3} + \frac{x^{3}}{3} + \frac{1}{3} x + K)$

Step 3.2.2.1.1.2.3

Combine $\frac{1}{3}$ and $x$ .

$y^{3} = 3 (\frac{x^{4}}{3} + \frac{x^{3}}{3} + \frac{x}{3} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{3} = 3 \frac{x^{4}}{3} + 3 \frac{x^{3}}{3} + 3 \frac{x}{3} + 3 K$

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.1.1

Cancel the common factor.

$y^{3} = 3 \frac{x^{4}}{3} + 3 \frac{x^{3}}{3} + 3 \frac{x}{3} + 3 K$

Step 3.2.2.1.3.1.2

Rewrite the expression.

$y^{3} = x^{4} + 3 \frac{x^{3}}{3} + 3 \frac{x}{3} + 3 K$

Step 3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.2.1

Cancel the common factor.

$y^{3} = x^{4} + 3 \frac{x^{3}}{3} + 3 \frac{x}{3} + 3 K$

Step 3.2.2.1.3.2.2

Rewrite the expression.

$y^{3} = x^{4} + x^{3} + 3 \frac{x}{3} + 3 K$

Step 3.2.2.1.3.3

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.3.1

Cancel the common factor.

$y^{3} = x^{4} + x^{3} + 3 \frac{x}{3} + 3 K$

Step 3.2.2.1.3.3.2

Rewrite the expression.

$y^{3} = x^{4} + x^{3} + x + 3 K$

Step 3.3

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

$y = \sqrt[3]{x^{4} + x^{3} + x + 3 K}$

Step 4

Simplify the constant of integration.

$y = \sqrt[3]{x^{4} + x^{3} + x + K}$