Calculus Examples

$4 y \frac{d y}{d x} = 5 x^{2}$

Step 1

Rewrite the equation.

$4 y d y = 5 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 4 y d y = \int 5 x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

$4 \int y d y = \int 5 x^{2} d x$

Step 2.2.2

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

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $4 (\frac{1}{2} y^{2} + C_{1})$ as $4 (\frac{1}{2}) y^{2} + C_{1}$ .

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

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Combine $4$ and $\frac{1}{2}$ .

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

Step 2.2.3.2.2

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

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Step 2.2.3.2.2.1

Factor $2$ out of $4$ .

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

Step 2.2.3.2.2.2

Cancel the common factors.

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Step 2.2.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.2.3.2.2.2.2

Cancel the common factor.

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

Step 2.2.3.2.2.2.3

Rewrite the expression.

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

Step 2.2.3.2.2.2.4

Divide $2$ by $1$ .

$2 y^{2} + C_{1} = \int 5 x^{2} d x$

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.

$2 y^{2} + C_{1} = 5 \int x^{2} d x$

Step 2.3.2

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

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

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

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

Step 2.3.3.2

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

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

Step 2.4

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

$2 y^{2} = \frac{5}{3} x^{3} + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{2} = \frac{5}{3} x^{3} + K$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{2} = \frac{5}{3} x^{3} + K$ by $2$ .

$\frac{2 y^{2}}{2} = \frac{\frac{5}{3} x^{3}}{2} + \frac{K}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{\frac{5}{3} x^{3}}{2} + \frac{K}{2}$

Step 3.1.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{\frac{5}{3} x^{3}}{2} + \frac{K}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

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

$y^{2} = \frac{\frac{5 x^{3}}{3}}{2} + \frac{K}{2}$

Step 3.1.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y^{2} = \frac{5 x^{3}}{3} \cdot \frac{1}{2} + \frac{K}{2}$

Step 3.1.3.1.3

Combine.

$y^{2} = \frac{5 x^{3} \cdot 1}{3 \cdot 2} + \frac{K}{2}$

Step 3.1.3.1.4

Multiply $5$ by $1$ .

$y^{2} = \frac{5 x^{3}}{3 \cdot 2} + \frac{K}{2}$

Step 3.1.3.1.5

Multiply $3$ by $2$ .

$y^{2} = \frac{5 x^{3}}{6} + \frac{K}{2}$

Step 3.2

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

$y = \pm \sqrt{\frac{5 x^{3}}{6} + \frac{K}{2}}$

Step 3.3

Simplify $\pm \sqrt{\frac{5 x^{3}}{6} + \frac{K}{2}}$ .

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Step 3.3.1

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

$y = \pm \sqrt{\frac{5 x^{3}}{6} + \frac{K}{2} \cdot \frac{3}{3}}$

Step 3.3.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.1

Multiply $\frac{K}{2}$ by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{5 x^{3}}{6} + \frac{K \cdot 3}{2 \cdot 3}}$

Step 3.3.2.2

Multiply $2$ by $3$ .

$y = \pm \sqrt{\frac{5 x^{3}}{6} + \frac{K \cdot 3}{6}}$

Step 3.3.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{5 x^{3} + K \cdot 3}{6}}$

Step 3.3.4

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

$y = \pm \sqrt{\frac{5 x^{3} + 3 K}{6}}$

Step 3.3.5

Rewrite $\sqrt{\frac{5 x^{3} + 3 K}{6}}$ as $\frac{\sqrt{5 x^{3} + 3 K}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{5 x^{3} + 3 K}}{\sqrt{6}}$

Step 3.3.6

Multiply $\frac{\sqrt{5 x^{3} + 3 K}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{5 x^{3} + 3 K}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 3.3.7

Combine and simplify the denominator.

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Step 3.3.7.1

Multiply $\frac{\sqrt{5 x^{3} + 3 K}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 3.3.7.2

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 3.3.7.3

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 3.3.7.4

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

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 3.3.7.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 3.3.7.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.3.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 3.3.7.6.2

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

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 3.3.7.6.3

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

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 3.3.7.6.4

Cancel the common factor of $2$ .

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Step 3.3.7.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 3.3.7.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{6^{1}}$

Step 3.3.7.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{5 x^{3} + 3 K} \sqrt{6}}{6}$

Step 3.3.8

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(5 x^{3} + 3 K) \cdot 6}}{6}$

Step 3.3.9

Reorder factors in $\pm \frac{\sqrt{(5 x^{3} + 3 K) \cdot 6}}{6}$ .

$y = \pm \frac{\sqrt{6 (5 x^{3} + 3 K)}}{6}$

Step 3.4

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

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Step 3.4.1

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

$y = \frac{\sqrt{6 (5 x^{3} + 3 K)}}{6}$

Step 3.4.2

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

$y = - \frac{\sqrt{6 (5 x^{3} + 3 K)}}{6}$

Step 3.4.3

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

$y = \frac{\sqrt{6 (5 x^{3} + 3 K)}}{6}$

$y = - \frac{\sqrt{6 (5 x^{3} + 3 K)}}{6}$

$y = \frac{\sqrt{6 (5 x^{3} + 3 K)}}{6}$

$y = - \frac{\sqrt{6 (5 x^{3} + 3 K)}}{6}$

$y = \frac{\sqrt{6 (5 x^{3} + 3 K)}}{6}$

$y = - \frac{\sqrt{6 (5 x^{3} + 3 K)}}{6}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{6 (5 x^{3} + K)}}{6}$

$y = - \frac{\sqrt{6 (5 x^{3} + K)}}{6}$