Calculus Examples

$d x + 3 x^{2} y^{2} d y = 0$

Step 1

Subtract $d x$ from both sides of the equation.

$3 x^{2} y^{2} d y = - d x$

Step 2

Multiply both sides by $\frac{1}{x^{2}}$ .

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

Combine $3$ and $\frac{1}{x^{2}}$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Simplify the answer.

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Step 4.2.3.1

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

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

Step 4.2.3.2

Simplify.

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Step 4.2.3.2.1

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

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

Step 4.2.3.2.2

Cancel the common factor of $3$ .

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Step 4.2.3.2.2.1

Cancel the common factor.

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

Step 4.2.3.2.2.2

Rewrite the expression.

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

Step 4.2.3.2.3

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

Apply basic rules of exponents.

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Step 4.3.2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.3.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.3

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

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

Step 4.3.4

Simplify the answer.

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Step 4.3.4.1

Rewrite $- (- x^{- 1} + C_{2})$ as $- - \frac{1}{x} + C_{2}$ .

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

Step 4.3.4.2

Simplify.

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Step 4.3.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.4.2.2

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

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

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

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

Step 5.2

Simplify $\sqrt[3]{\frac{1}{x} + K}$ .

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Step 5.2.1

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

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

Step 5.2.2

Combine the numerators over the common denominator.

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

Step 5.2.3

Rewrite $\sqrt[3]{\frac{1 + K x}{x}}$ as $\frac{\sqrt[3]{1 + K x}}{\sqrt[3]{x}}$ .

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

Step 5.2.4

Multiply $\frac{\sqrt[3]{1 + K x}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

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

Step 5.2.5

Combine and simplify the denominator.

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Step 5.2.5.1

Multiply $\frac{\sqrt[3]{1 + K x}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

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

Step 5.2.5.2

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

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

Step 5.2.5.3

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

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

Step 5.2.5.4

Add $1$ and $2$ .

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

Step 5.2.5.5

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

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Step 5.2.5.5.1

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

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

Step 5.2.5.5.2

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

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

Step 5.2.5.5.3

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

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

Step 5.2.5.5.4

Cancel the common factor of $3$ .

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Step 5.2.5.5.4.1

Cancel the common factor.

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

Step 5.2.5.5.4.2

Rewrite the expression.

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

Step 5.2.5.5.5

Simplify.

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

Step 5.2.6

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

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

Step 5.2.7

Combine using the product rule for radicals.

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

Step 5.2.8

Reorder factors in $\frac{\sqrt[3]{(1 + K x) x^{2}}}{x}$ .

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