Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

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.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

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

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

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

Step 2.4

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

$\frac{1}{3} y^{3} = \frac{1}{3} (\frac{1}{2} x^{2} - x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} y^{3}) = 3 (\frac{1}{3} (\frac{1}{2} x^{2} - x) + K)$

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 $y^{3}$ .

$3 \frac{y^{3}}{3} = 3 (\frac{1}{3} (\frac{1}{2} x^{2} - x) + K)$

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.

$3 \frac{y^{3}}{3} = 3 (\frac{1}{3} (\frac{1}{2} x^{2} - x) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (\frac{1}{3} (\frac{1}{2} x^{2} - x) + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (\frac{1}{3} (\frac{1}{2} x^{2} - x) + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$y^{3} = 3 (\frac{1}{3} (\frac{x^{2}}{2} - x) + K)$

Step 3.2.2.1.1.2

Apply the distributive property.

$y^{3} = 3 (\frac{1}{3} \cdot \frac{x^{2}}{2} + \frac{1}{3} (- x) + K)$

Step 3.2.2.1.1.3

Combine.

$y^{3} = 3 (\frac{1 x^{2}}{3 \cdot 2} + \frac{1}{3} (- x) + K)$

Step 3.2.2.1.1.4

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

$y^{3} = 3 (\frac{1 x^{2}}{3 \cdot 2} - \frac{x}{3} + K)$

Step 3.2.2.1.1.5

Simplify each term.

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Step 3.2.2.1.1.5.1

Multiply $x^{2}$ by $1$ .

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

Step 3.2.2.1.1.5.2

Multiply $3$ by $2$ .

$y^{3} = 3 (\frac{x^{2}}{6} - \frac{x}{3} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{3} = 3 \frac{x^{2}}{6} + 3 (- \frac{x}{3}) + 3 K$

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

Factor $3$ out of $6$ .

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

Step 3.2.2.1.3.1.2

Cancel the common factor.

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

Step 3.2.2.1.3.1.3

Rewrite the expression.

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

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

Move the leading negative in $- \frac{x}{3}$ into the numerator.

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

Step 3.2.2.1.3.2.2

Cancel the common factor.

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

Step 3.2.2.1.3.2.3

Rewrite the expression.

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

Step 3.3

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

$y = \sqrt[3]{\frac{x^{2}}{2} - x + 3 K}$

Step 3.4

Simplify $\sqrt[3]{\frac{x^{2}}{2} - x + 3 K}$ .

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Step 3.4.1

To write $- x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt[3]{\frac{x^{2}}{2} - x \cdot \frac{2}{2} + 3 K}$

Step 3.4.2

Simplify terms.

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Step 3.4.2.1

Combine $- x$ and $\frac{2}{2}$ .

$y = \sqrt[3]{\frac{x^{2}}{2} + \frac{- x \cdot 2}{2} + 3 K}$

Step 3.4.2.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x^{2} - x \cdot 2}{2} + 3 K}$

Step 3.4.3

Simplify the numerator.

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Step 3.4.3.1

Factor $x$ out of $x^{2} - x \cdot 2$ .

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Step 3.4.3.1.1

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

$y = \sqrt[3]{\frac{x \cdot x - x \cdot 2}{2} + 3 K}$

Step 3.4.3.1.2

Factor $x$ out of $- x \cdot 2$ .

$y = \sqrt[3]{\frac{x \cdot x + x (- 1 \cdot 2)}{2} + 3 K}$

Step 3.4.3.1.3

Factor $x$ out of $x \cdot x + x (- 1 \cdot 2)$ .

$y = \sqrt[3]{\frac{x (x - 1 \cdot 2)}{2} + 3 K}$

Step 3.4.3.2

Multiply $- 1$ by $2$ .

$y = \sqrt[3]{\frac{x (x - 2)}{2} + 3 K}$

Step 3.4.4

To write $3 K$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt[3]{\frac{x (x - 2)}{2} + 3 K \cdot \frac{2}{2}}$

Step 3.4.5

Simplify terms.

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Step 3.4.5.1

Combine $3 K$ and $\frac{2}{2}$ .

$y = \sqrt[3]{\frac{x (x - 2)}{2} + \frac{3 K \cdot 2}{2}}$

Step 3.4.5.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x (x - 2) + 3 K \cdot 2}{2}}$

Step 3.4.6

Simplify the numerator.

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Step 3.4.6.1

Apply the distributive property.

$y = \sqrt[3]{\frac{x \cdot x + x \cdot - 2 + 3 K \cdot 2}{2}}$

Step 3.4.6.2

Multiply $x$ by $x$ .

$y = \sqrt[3]{\frac{x^{2} + x \cdot - 2 + 3 K \cdot 2}{2}}$

Step 3.4.6.3

Move $- 2$ to the left of $x$ .

$y = \sqrt[3]{\frac{x^{2} - 2 \cdot x + 3 K \cdot 2}{2}}$

Step 3.4.6.4

Multiply $2$ by $3$ .

$y = \sqrt[3]{\frac{x^{2} - 2 x + 6 K}{2}}$

Step 3.4.7

Rewrite $\sqrt[3]{\frac{x^{2} - 2 x + 6 K}{2}}$ as $\frac{\sqrt[3]{x^{2} - 2 x + 6 K}}{\sqrt[3]{2}}$ .

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K}}{\sqrt[3]{2}}$

Step 3.4.8

Multiply $\frac{\sqrt[3]{x^{2} - 2 x + 6 K}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$

Step 3.4.9

Combine and simplify the denominator.

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Step 3.4.9.1

Multiply $\frac{\sqrt[3]{x^{2} - 2 x + 6 K}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{\sqrt[3]{2} {\sqrt[3]{2}}^{2}}$

Step 3.4.9.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

Step 3.4.9.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 3.4.9.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{3}}$

Step 3.4.9.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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Step 3.4.9.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 3.4.9.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Step 3.4.9.5.3

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

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.4.9.5.4

Cancel the common factor of $3$ .

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Step 3.4.9.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{2^{\frac{3}{3}}}$

Step 3.4.9.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{2^{1}}$

Step 3.4.9.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} {\sqrt[3]{2}}^{2}}{2}$

Step 3.4.10

Simplify the numerator.

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Step 3.4.10.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} \sqrt[3]{2^{2}}}{2}$

Step 3.4.10.2

Raise $2$ to the power of $2$ .

$y = \frac{\sqrt[3]{x^{2} - 2 x + 6 K} \sqrt[3]{4}}{2}$

Step 3.4.11

Simplify with factoring out.

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Step 3.4.11.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x^{2} - 2 x + 6 K) \cdot 4}}{2}$

Step 3.4.11.2

Reorder factors in $\frac{\sqrt[3]{(x^{2} - 2 x + 6 K) \cdot 4}}{2}$ .

$y = \frac{\sqrt[3]{4 (x^{2} - 2 x + 6 K)}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{4 (x^{2} - 2 x + K)}}{2}$