Calculus Examples

$5 x^{2} d x + 7 y^{3} d y = 0$

Step 1

Subtract $5 x^{2} d x$ from both sides of the equation.

$7 y^{3} d y = - 5 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 7 y^{3} d y = \int - 5 x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $7$ is constant with respect to $y$ , move $7$ out of the integral.

$7 \int y^{3} d y = \int - 5 x^{2} d x$

Step 2.2.2

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

$7 (\frac{1}{4} y^{4} + C_{1}) = \int - 5 x^{2} d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $7 (\frac{1}{4} y^{4} + C_{1})$ as $7 (\frac{1}{4}) y^{4} + C_{1}$ .

$7 (\frac{1}{4}) y^{4} + C_{1} = \int - 5 x^{2} d x$

Step 2.2.3.2

Combine $7$ and $\frac{1}{4}$ .

$\frac{7}{4} y^{4} + C_{1} = \int - 5 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$\frac{7}{4} y^{4} + C_{1} = - 5 \int x^{2} d x$

Step 2.3.2

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

$\frac{7}{4} y^{4} + C_{1} = - 5 (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $- 5 (\frac{1}{3} x^{3} + C_{2})$ as $- 5 (\frac{1}{3}) x^{3} + C_{2}$ .

$\frac{7}{4} y^{4} + C_{1} = - 5 (\frac{1}{3}) x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $- 5$ and $\frac{1}{3}$ .

$\frac{7}{4} y^{4} + C_{1} = \frac{- 5}{3} x^{3} + C_{2}$

Step 2.3.3.2.2

Move the negative in front of the fraction.

$\frac{7}{4} y^{4} + C_{1} = - \frac{5}{3} x^{3} + C_{2}$

Step 2.4

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

$\frac{7}{4} y^{4} = - \frac{5}{3} x^{3} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $\frac{4}{7}$ .

$\frac{4}{7} (\frac{7}{4} y^{4}) = \frac{4}{7} (- \frac{5}{3} x^{3} + K)$

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 $\frac{4}{7} (\frac{7}{4} y^{4})$ .

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Step 3.2.1.1.1

Combine $\frac{7}{4}$ and $y^{4}$ .

$\frac{4}{7} \cdot \frac{7 y^{4}}{4} = \frac{4}{7} (- \frac{5}{3} x^{3} + K)$

Step 3.2.1.1.2

Combine.

$\frac{4 (7 y^{4})}{7 \cdot 4} = \frac{4}{7} (- \frac{5}{3} x^{3} + K)$

Step 3.2.1.1.3

Cancel the common factor of $4$ .

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Step 3.2.1.1.3.1

Cancel the common factor.

$\frac{4 (7 y^{4})}{7 \cdot 4} = \frac{4}{7} (- \frac{5}{3} x^{3} + K)$

Step 3.2.1.1.3.2

Rewrite the expression.

$\frac{7 y^{4}}{7} = \frac{4}{7} (- \frac{5}{3} x^{3} + K)$

Step 3.2.1.1.4

Cancel the common factor of $7$ .

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Step 3.2.1.1.4.1

Cancel the common factor.

$\frac{7 y^{4}}{7} = \frac{4}{7} (- \frac{5}{3} x^{3} + K)$

Step 3.2.1.1.4.2

Divide $y^{4}$ by $1$ .

$y^{4} = \frac{4}{7} (- \frac{5}{3} x^{3} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{4}{7} (- \frac{5}{3} x^{3} + K)$ .

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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 $x^{3}$ and $\frac{5}{3}$ .

$y^{4} = \frac{4}{7} (- \frac{x^{3} \cdot 5}{3} + K)$

Step 3.2.2.1.1.2

Move $5$ to the left of $x^{3}$ .

$y^{4} = \frac{4}{7} (- \frac{5 x^{3}}{3} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{4} = \frac{4}{7} (- \frac{5 x^{3}}{3}) + \frac{4}{7} K$

Step 3.2.2.1.3

Multiply $\frac{4}{7} (- \frac{5 x^{3}}{3})$ .

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Step 3.2.2.1.3.1

Multiply $\frac{4}{7}$ by $\frac{5 x^{3}}{3}$ .

$y^{4} = - \frac{4 (5 x^{3})}{7 \cdot 3} + \frac{4}{7} K$

Step 3.2.2.1.3.2

Multiply $5$ by $4$ .

$y^{4} = - \frac{20 x^{3}}{7 \cdot 3} + \frac{4}{7} K$

Step 3.2.2.1.3.3

Multiply $7$ by $3$ .

$y^{4} = - \frac{20 x^{3}}{21} + \frac{4}{7} K$

Step 3.2.2.1.4

Combine $\frac{4}{7}$ and $K$ .

$y^{4} = - \frac{20 x^{3}}{21} + \frac{4 K}{7}$

Step 3.3

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

$y = \pm \sqrt[4]{- \frac{20 x^{3}}{21} + \frac{4 K}{7}}$

Step 3.4

Simplify $\pm \sqrt[4]{- \frac{20 x^{3}}{21} + \frac{4 K}{7}}$ .

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Step 3.4.1

To write $\frac{4 K}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt[4]{- \frac{20 x^{3}}{21} + \frac{4 K}{7} \cdot \frac{3}{3}}$

Step 3.4.2

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.1

Multiply $\frac{4 K}{7}$ by $\frac{3}{3}$ .

$y = \pm \sqrt[4]{- \frac{20 x^{3}}{21} + \frac{4 K \cdot 3}{7 \cdot 3}}$

Step 3.4.2.2

Multiply $7$ by $3$ .

$y = \pm \sqrt[4]{- \frac{20 x^{3}}{21} + \frac{4 K \cdot 3}{21}}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt[4]{\frac{- 20 x^{3} + 4 K \cdot 3}{21}}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Factor $4$ out of $- 20 x^{3} + 4 K \cdot 3$ .

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Step 3.4.4.1.1

Factor $4$ out of $- 20 x^{3}$ .

$y = \pm \sqrt[4]{\frac{4 (- 5 x^{3}) + 4 K \cdot 3}{21}}$

Step 3.4.4.1.2

Factor $4$ out of $4 K \cdot 3$ .

$y = \pm \sqrt[4]{\frac{4 (- 5 x^{3}) + 4 (K \cdot 3)}{21}}$

Step 3.4.4.1.3

Factor $4$ out of $4 (- 5 x^{3}) + 4 (K \cdot 3)$ .

$y = \pm \sqrt[4]{\frac{4 (- 5 x^{3} + K \cdot 3)}{21}}$

Step 3.4.4.2

Move $3$ to the left of $K$ .

$y = \pm \sqrt[4]{\frac{4 (- 5 x^{3} + 3 K)}{21}}$

Step 3.4.5

Rewrite $\sqrt[4]{\frac{4 (- 5 x^{3} + 3 K)}{21}}$ as $\frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)}}{\sqrt[4]{21}}$ .

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)}}{\sqrt[4]{21}}$

Step 3.4.6

Multiply $\frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)}}{\sqrt[4]{21}}$ by $\frac{{\sqrt[4]{21}}^{3}}{{\sqrt[4]{21}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)}}{\sqrt[4]{21}} \cdot \frac{{\sqrt[4]{21}}^{3}}{{\sqrt[4]{21}}^{3}}$

Step 3.4.7

Combine and simplify the denominator.

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Step 3.4.7.1

Multiply $\frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)}}{\sqrt[4]{21}}$ by $\frac{{\sqrt[4]{21}}^{3}}{{\sqrt[4]{21}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{\sqrt[4]{21} {\sqrt[4]{21}}^{3}}$

Step 3.4.7.2

Raise $\sqrt[4]{21}$ to the power of $1$ .

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{{\sqrt[4]{21}}^{1} {\sqrt[4]{21}}^{3}}$

Step 3.4.7.3

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

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{{\sqrt[4]{21}}^{1 + 3}}$

Step 3.4.7.4

Add $1$ and $3$ .

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{{\sqrt[4]{21}}^{4}}$

Step 3.4.7.5

Rewrite ${\sqrt[4]{21}}^{4}$ as $21$ .

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Step 3.4.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{21}$ as $21^{\frac{1}{4}}$ .

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{{(21^{\frac{1}{4}})}^{4}}$

Step 3.4.7.5.2

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

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{21^{\frac{1}{4} \cdot 4}}$

Step 3.4.7.5.3

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

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{21^{\frac{4}{4}}}$

Step 3.4.7.5.4

Cancel the common factor of $4$ .

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Step 3.4.7.5.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{21^{\frac{4}{4}}}$

Step 3.4.7.5.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{21^{1}}$

Step 3.4.7.5.5

Evaluate the exponent.

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} {\sqrt[4]{21}}^{3}}{21}$

Step 3.4.8

Simplify the numerator.

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Step 3.4.8.1

Rewrite ${\sqrt[4]{21}}^{3}$ as $\sqrt[4]{21^{3}}$ .

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} \sqrt[4]{21^{3}}}{21}$

Step 3.4.8.2

Raise $21$ to the power of $3$ .

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K)} \sqrt[4]{9261}}{21}$

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt[4]{4 (- 5 x^{3} + 3 K) \cdot 9261}}{21}$

Step 3.4.9.2

Multiply $9261$ by $4$ .

$y = \pm \frac{\sqrt[4]{37044 (- 5 x^{3} + 3 K)}}{21}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt[4]{37044 (- 5 x^{3} + 3 K)}}{21}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt[4]{37044 (- 5 x^{3} + 3 K)}}{21}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt[4]{37044 (- 5 x^{3} + 3 K)}}{21}$

$y = - \frac{\sqrt[4]{37044 (- 5 x^{3} + 3 K)}}{21}$

$y = \frac{\sqrt[4]{37044 (- 5 x^{3} + 3 K)}}{21}$

$y = - \frac{\sqrt[4]{37044 (- 5 x^{3} + 3 K)}}{21}$

$y = \frac{\sqrt[4]{37044 (- 5 x^{3} + 3 K)}}{21}$

$y = - \frac{\sqrt[4]{37044 (- 5 x^{3} + 3 K)}}{21}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[4]{37044 (- 5 x^{3} + K)}}{21}$

$y = - \frac{\sqrt[4]{37044 (- 5 x^{3} + K)}}{21}$