Calculus Examples

$3 \frac{d y}{d x} = (4 x^{3} - 1) y^{4}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y^{4}}$ .

$\frac{1}{y^{4}} (3 \frac{d y}{d x}) = \frac{1}{y^{4}} ((4 x^{3} - 1) y^{4})$

Step 1.2

Cancel the common factor of $y^{4}$ .

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Step 1.2.1

Factor $y^{4}$ out of $(4 x^{3} - 1) y^{4}$ .

$\frac{1}{y^{4}} (3 \frac{d y}{d x}) = \frac{1}{y^{4}} (y^{4} (4 x^{3} - 1))$

Step 1.2.2

Cancel the common factor.

$\frac{1}{y^{4}} (3 \frac{d y}{d x}) = \frac{1}{y^{4}} (y^{4} (4 x^{3} - 1))$

Step 1.2.3

Rewrite the expression.

$\frac{1}{y^{4}} (3 \frac{d y}{d x}) = 4 x^{3} - 1$

Step 1.3

Remove unnecessary parentheses.

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

Step 1.4

Rewrite the equation.

$\frac{1}{y^{4}} \cdot 3 d y = (4 x^{3} - 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{4}} \cdot 3 d y = \int 4 x^{3} - 1 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Apply basic rules of exponents.

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Step 2.2.3.1

Move $y^{4}$ out of the denominator by raising it to the $- 1$ power.

$3 \int {(y^{4})}^{- 1} d y = \int 4 x^{3} - 1 d x$

Step 2.2.3.2

Multiply the exponents in ${(y^{4})}^{- 1}$ .

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Step 2.2.3.2.1

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

$3 \int y^{4 \cdot - 1} d y = \int 4 x^{3} - 1 d x$

Step 2.2.3.2.2

Multiply $4$ by $- 1$ .

$3 \int y^{- 4} d y = \int 4 x^{3} - 1 d x$

Step 2.2.4

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

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

Step 2.2.5

Simplify the answer.

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Step 2.2.5.1

Simplify.

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Step 2.2.5.1.1

Combine $y^{- 3}$ and $\frac{1}{3}$ .

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

Step 2.2.5.1.2

Move $y^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2.5.2

Simplify.

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

Step 2.2.5.3

Simplify.

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Step 2.2.5.3.1

Multiply $- 1$ by $3$ .

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

Step 2.2.5.3.2

Combine $- 3$ and $\frac{1}{3 y^{3}}$ .

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

Step 2.2.5.3.3

Cancel the common factor of $- 3$ and $3$ .

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Step 2.2.5.3.3.1

Factor $3$ out of $- 3$ .

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

Step 2.2.5.3.3.2

Cancel the common factors.

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Step 2.2.5.3.3.2.1

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

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

Step 2.2.5.3.3.2.2

Cancel the common factor.

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

Step 2.2.5.3.3.2.3

Rewrite the expression.

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

Step 2.2.5.3.4

Move the negative in front of the fraction.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{3}, 1, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$y^{3}$

Step 3.2

Multiply each term in $- \frac{1}{y^{3}} = x^{4} - x + K$ by $y^{3}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $- \frac{1}{y^{3}} = x^{4} - x + K$ by $y^{3}$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $y^{3}$ .

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Step 3.2.2.1.1

Move the leading negative in $- \frac{1}{y^{3}}$ into the numerator.

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

Step 3.2.2.1.2

Cancel the common factor.

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

Step 3.2.2.1.3

Rewrite the expression.

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

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $x^{4} y^{3} - x y^{3} + K y^{3} = - 1$ .

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

Step 3.3.2

Factor $y^{3}$ out of $x^{4} y^{3} - x y^{3} + K y^{3}$ .

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Step 3.3.2.1

Factor $y^{3}$ out of $x^{4} y^{3}$ .

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

Step 3.3.2.2

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

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

Step 3.3.2.3

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

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

Step 3.3.2.4

Factor $y^{3}$ out of $y^{3} x^{4} + y^{3} (- x)$ .

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

Step 3.3.2.5

Factor $y^{3}$ out of $y^{3} (x^{4} - x) + y^{3} K$ .

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

Step 3.3.3

Divide each term in $y^{3} (x^{4} - x + K) = - 1$ by $x^{4} - x + K$ and simplify.

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Step 3.3.3.1

Divide each term in $y^{3} (x^{4} - x + K) = - 1$ by $x^{4} - x + K$ .

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

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $x^{4} - x + K$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.1.2

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

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

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

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

Step 3.3.4

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

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

Step 3.3.5

Simplify $\sqrt[3]{- \frac{1}{x^{4} - x + K}}$ .

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Step 3.3.5.1

Rewrite $- \frac{1}{x^{4} - x + K}$ as ${({(- 1)}^{3})}^{3} \frac{1}{x^{4} - x + K}$ .

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Step 3.3.5.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 3.3.5.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 3.3.5.2

Pull terms out from under the radical.

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

Step 3.3.5.3

Raise $- 1$ to the power of $3$ .

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

Step 3.3.5.4

Rewrite $\sqrt[3]{\frac{1}{x^{4} - x + K}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{x^{4} - x + K}}$ .

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

Step 3.3.5.5

Any root of $1$ is $1$ .

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

Step 3.3.5.6

Multiply $\frac{1}{\sqrt[3]{x^{4} - x + K}}$ by $\frac{{\sqrt[3]{x^{4} - x + K}}^{2}}{{\sqrt[3]{x^{4} - x + K}}^{2}}$ .

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

Step 3.3.5.7

Combine and simplify the denominator.

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Step 3.3.5.7.1

Multiply $\frac{1}{\sqrt[3]{x^{4} - x + K}}$ by $\frac{{\sqrt[3]{x^{4} - x + K}}^{2}}{{\sqrt[3]{x^{4} - x + K}}^{2}}$ .

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

Step 3.3.5.7.2

Raise $\sqrt[3]{x^{4} - x + K}$ to the power of $1$ .

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

Step 3.3.5.7.3

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

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

Step 3.3.5.7.4

Add $1$ and $2$ .

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

Step 3.3.5.7.5

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

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Step 3.3.5.7.5.1

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

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

Step 3.3.5.7.5.2

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

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

Step 3.3.5.7.5.3

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

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

Step 3.3.5.7.5.4

Cancel the common factor of $3$ .

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Step 3.3.5.7.5.4.1

Cancel the common factor.

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

Step 3.3.5.7.5.4.2

Rewrite the expression.

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

Step 3.3.5.7.5.5

Simplify.

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

Step 3.3.5.8

Rewrite ${\sqrt[3]{x^{4} - x + K}}^{2}$ as $\sqrt[3]{{(x^{4} - x + K)}^{2}}$ .

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