Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Flip sides to get $\frac{d y}{d x}$ on the left side.

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

Step 1.2

Multiply both sides by $y$ .

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

Step 1.3

Cancel the common factor of $y$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Remove unnecessary parentheses.

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

Step 1.5

Rewrite the equation.

$y \cdot 4 y d y = x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y \cdot 4 y d y = \int x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Move $4$ to the left of $y$ .

$\int 4 \cdot y \cdot y d y = \int x^{2} d x$

Step 2.2.1.2

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

$\int 4 (y^{1} y) d y = \int x^{2} d x$

Step 2.2.1.3

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

$\int 4 (y^{1} y^{1}) d y = \int x^{2} d x$

Step 2.2.1.4

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

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

Step 2.2.1.5

Add $1$ and $1$ .

$\int 4 y^{2} d y = \int x^{2} d x$

Step 2.2.2

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

$4 \int y^{2} d y = \int x^{2} d x$

Step 2.2.3

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

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

Step 2.2.4

Simplify the answer.

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Step 2.2.4.1

Rewrite $4 (\frac{1}{3} y^{3} + C_{1})$ as $4 (\frac{1}{3}) y^{3} + C_{1}$ .

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

Step 2.2.4.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $\frac{3}{4}$ .

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

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 $\frac{3}{4} (\frac{4}{3} y^{3})$ .

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Step 3.2.1.1.1

Combine $\frac{4}{3}$ and $y^{3}$ .

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

Step 3.2.1.1.2

Combine.

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

Step 3.2.1.1.3

Cancel the common factor of $3$ .

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Step 3.2.1.1.3.1

Cancel the common factor.

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

Step 3.2.1.1.3.2

Rewrite the expression.

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

Step 3.2.1.1.4

Cancel the common factor of $4$ .

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Step 3.2.1.1.4.1

Cancel the common factor.

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

Step 3.2.1.1.4.2

Divide $y^{3}$ by $1$ .

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

Step 3.2.2.1.3

Combine.

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

Step 3.2.2.1.4

Combine $\frac{3}{4}$ and $K$ .

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

Step 3.2.2.1.5

Cancel the common factor of $3$ .

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Step 3.2.2.1.5.1

Cancel the common factor.

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

Step 3.2.2.1.5.2

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Simplify $\sqrt[3]{\frac{x^{3}}{4} + \frac{3 K}{4}}$ .

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Step 3.4.1

Combine the numerators over the common denominator.

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

Step 3.4.2

Rewrite $\sqrt[3]{\frac{x^{3} + 3 K}{4}}$ as $\frac{\sqrt[3]{x^{3} + 3 K}}{\sqrt[3]{4}}$ .

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

Step 3.4.3

Multiply $\frac{\sqrt[3]{x^{3} + 3 K}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

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

Step 3.4.4

Combine and simplify the denominator.

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Step 3.4.4.1

Multiply $\frac{\sqrt[3]{x^{3} + 3 K}}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

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

Step 3.4.4.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}}$

Step 3.4.4.3

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

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}}$

Step 3.4.4.4

Add $1$ and $2$ .

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

Step 3.4.4.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 3.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}}$

Step 3.4.4.5.2

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

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}}$

Step 3.4.4.5.3

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

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

Step 3.4.4.5.4

Cancel the common factor of $3$ .

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Step 3.4.4.5.4.1

Cancel the common factor.

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

Step 3.4.4.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x^{3} + 3 K} {\sqrt[3]{4}}^{2}}{4^{1}}$

Step 3.4.4.5.5

Evaluate the exponent.

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

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

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

Step 3.4.5.2

Raise $4$ to the power of $2$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K} \sqrt[3]{16}}{4}$

Step 3.4.5.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 3.4.5.3.1

Factor $8$ out of $16$ .

$y = \frac{\sqrt[3]{x^{3} + 3 K} \sqrt[3]{8 (2)}}{4}$

Step 3.4.5.3.2

Rewrite $8$ as $2^{3}$ .

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

Step 3.4.5.4

Pull terms out from under the radical.

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

Step 3.4.5.5

Combine using the product rule for radicals.

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

Step 3.4.6

Reduce the expression by cancelling the common factors.

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Step 3.4.6.1

Cancel the common factor of $2$ and $4$ .

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Step 3.4.6.1.1

Factor $2$ out of $2 \sqrt[3]{(x^{3} + 3 K) \cdot 2}$ .

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

Step 3.4.6.1.2

Cancel the common factors.

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Step 3.4.6.1.2.1

Factor $2$ out of $4$ .

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

Step 3.4.6.1.2.2

Cancel the common factor.

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

Step 3.4.6.1.2.3

Rewrite the expression.

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

Step 3.4.6.2

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

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

Step 4

Simplify the constant of integration.

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