Calculus Examples

$8 y \frac{d y}{d x} = \frac{12}{{(3 x + 2)}^{2}}$

Step 1

Rewrite the equation.

$8 y d y = \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 8 y d y = \int \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $8$ is constant with respect to $y$ , move $8$ out of the integral.

$8 \int y d y = \int \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2.2.2

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

$8 (\frac{1}{2} y^{2} + C_{1}) = \int \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $8 (\frac{1}{2} y^{2} + C_{1})$ as $8 (\frac{1}{2}) y^{2} + C_{1}$ .

$8 (\frac{1}{2}) y^{2} + C_{1} = \int \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Combine $8$ and $\frac{1}{2}$ .

$\frac{8}{2} y^{2} + C_{1} = \int \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2.2.3.2.2

Cancel the common factor of $8$ and $2$ .

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Step 2.2.3.2.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4}{2} y^{2} + C_{1} = \int \frac{12}{{(3 x + 2)}^{2}} d x$

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 $2$ .

$\frac{2 \cdot 4}{2 (1)} y^{2} + C_{1} = \int \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2.2.3.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 1} y^{2} + C_{1} = \int \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2.2.3.2.2.2.3

Rewrite the expression.

$\frac{4}{1} y^{2} + C_{1} = \int \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2.2.3.2.2.2.4

Divide $4$ by $1$ .

$4 y^{2} + C_{1} = \int \frac{12}{{(3 x + 2)}^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$4 y^{2} + C_{1} = 12 \int \frac{1}{{(3 x + 2)}^{2}} d x$

Step 2.3.2

Let $u = 3 x + 2$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 3 x + 2$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $3 x + 2$ .

$\frac{d}{d x} [3 x + 2]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $3 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$

Step 2.3.2.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 2.3.2.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 2.3.2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1 + \frac{d}{d x} [2]$

Step 2.3.2.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [2]$

Step 2.3.2.1.4

Differentiate using the Constant Rule.

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Step 2.3.2.1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$3 + 0$

Step 2.3.2.1.4.2

Add $3$ and $0$ .

$3$

Step 2.3.2.2

Rewrite the problem using $u$ and $d u$ .

$4 y^{2} + C_{1} = 12 \int \frac{1}{u^{2}} \cdot \frac{1}{3} d u$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $\frac{1}{u^{2}}$ by $\frac{1}{3}$ .

$4 y^{2} + C_{1} = 12 \int \frac{1}{u^{2} \cdot 3} d u$

Step 2.3.3.2

Move $3$ to the left of $u^{2}$ .

$4 y^{2} + C_{1} = 12 \int \frac{1}{3 u^{2}} d u$

Step 2.3.4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$4 y^{2} + C_{1} = 12 (\frac{1}{3} \int \frac{1}{u^{2}} d u)$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Simplify.

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Step 2.3.5.1.1

Combine $\frac{1}{3}$ and $12$ .

$4 y^{2} + C_{1} = \frac{12}{3} \int \frac{1}{u^{2}} d u$

Step 2.3.5.1.2

Cancel the common factor of $12$ and $3$ .

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Step 2.3.5.1.2.1

Factor $3$ out of $12$ .

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

Step 2.3.5.1.2.2

Cancel the common factors.

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Step 2.3.5.1.2.2.1

Factor $3$ out of $3$ .

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

Step 2.3.5.1.2.2.2

Cancel the common factor.

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

Step 2.3.5.1.2.2.3

Rewrite the expression.

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

Step 2.3.5.1.2.2.4

Divide $4$ by $1$ .

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

Step 2.3.5.2

Apply basic rules of exponents.

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

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

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Step 2.3.5.2.2.1

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

$4 y^{2} + C_{1} = 4 \int u^{2 \cdot - 1} d u$

Step 2.3.5.2.2.2

Multiply $2$ by $- 1$ .

$4 y^{2} + C_{1} = 4 \int u^{- 2} d u$

Step 2.3.6

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

$4 y^{2} + C_{1} = 4 (- u^{- 1} + C_{2})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Multiply $- 1$ by $4$ .

$4 y^{2} + C_{1} = - 4 \frac{1}{u} + C_{2}$

Step 2.3.7.2.2

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

$4 y^{2} + C_{1} = \frac{- 4}{u} + C_{2}$

Step 2.3.7.2.3

Move the negative in front of the fraction.

$4 y^{2} + C_{1} = - \frac{4}{u} + C_{2}$

Step 2.3.8

Replace all occurrences of $u$ with $3 x + 2$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $4 y^{2} = - \frac{4}{3 x + 2} + K$ by $4$ and simplify.

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Step 3.1.1

Divide each term in $4 y^{2} = - \frac{4}{3 x + 2} + K$ by $4$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $4$ .

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Step 3.1.2.1.1

Cancel the common factor.

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

Step 3.1.2.1.2

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

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Combine the numerators over the common denominator.

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

Step 3.1.3.2

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

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

Step 3.1.3.3

Combine the numerators over the common denominator.

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

Step 3.1.3.4

Simplify the numerator.

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Step 3.1.3.4.1

Apply the distributive property.

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

Step 3.1.3.4.2

Rewrite using the commutative property of multiplication.

$y^{2} = \frac{\frac{- 4 + 3 K x + K \cdot 2}{3 x + 2}}{4}$

Step 3.1.3.4.3

Move $2$ to the left of $K$ .

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

Step 3.1.3.5

Simplify with factoring out.

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Step 3.1.3.5.1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 3.1.3.5.2

Factor $- 1$ out of $3 K x$ .

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

Step 3.1.3.5.3

Factor $- 1$ out of $- 1 (4) - (- 3 K x)$ .

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

Step 3.1.3.5.4

Factor $- 1$ out of $2 K$ .

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

Step 3.1.3.5.5

Factor $- 1$ out of $- 1 (4 - 3 K x) - (- 2 K)$ .

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

Step 3.1.3.5.6

Move the negative in front of the fraction.

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

Step 3.1.3.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3.7

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

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

Step 3.2

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

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

Step 3.3

Simplify $\pm \sqrt{- \frac{4 - 3 K x - 2 K}{4 (3 x + 2)}}$ .

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Step 3.3.1

Rewrite $- \frac{4 - 3 K x - 2 K}{4 (3 x + 2)}$ as ${(\frac{1^{2}}{2})}^{2} \cdot (- \frac{4 - 3 K x - 2 K}{3 x + 2})$ .

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Step 3.3.1.1

Factor the perfect power $1^{2}$ out of $4 - 3 K x - 2 K$ .

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

Step 3.3.1.2

Factor the perfect power $2^{2}$ out of $4 (3 x + 2)$ .

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

Step 3.3.1.3

Rearrange the fraction $\frac{1^{2} (4 - 3 K x - 2 K)}{2^{2} (3 x + 2)}$ .

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

Step 3.3.1.4

Reorder $- 1$ and ${(\frac{1}{2})}^{2}$ .

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

Step 3.3.1.5

Rewrite $1$ as $1^{2}$ .

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

Step 3.3.1.6

Add parentheses.

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

Step 3.3.2

Pull terms out from under the radical.

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

Step 3.3.3

One to any power is one.

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

Step 3.3.4

Combine $\frac{1}{2}$ and $\sqrt{- \frac{4 - 3 K x - 2 K}{3 x + 2}}$ .

$y = \pm \frac{\sqrt{- \frac{4 - 3 K x - 2 K}{3 x + 2}}}{2}$

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.

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

Step 3.4.2

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

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

Step 3.4.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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