Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by ${(2 y + 1)}^{2}$ .

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

Step 1.2

Cancel the common factor of ${(2 y + 1)}^{2}$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

${(2 y + 1)}^{2} d y = 4 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Let $u = 2 y + 1$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $2 y + 1$ .

$\frac{d}{d y} [2 y + 1]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $2 y + 1$ with respect to $y$ is $\frac{d}{d y} [2 y] + \frac{d}{d y} [1]$ .

$\frac{d}{d y} [2 y] + \frac{d}{d y} [1]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d y} [2 y]$ .

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Step 2.2.1.1.3.1

Since $2$ is constant with respect to $y$ , the derivative of $2 y$ with respect to $y$ is $2 \frac{d}{d y} [y]$ .

$2 \frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 2.2.1.1.3.2

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

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

Step 2.2.1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d y} [1]$

Step 2.2.1.1.4

Differentiate using the Constant Rule.

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Step 2.2.1.1.4.1

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$2 + 0$

Step 2.2.1.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2.1.2

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

$\int u^{2} \frac{1}{2} d u = \int 4 d x$

Step 2.2.2

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

$\int \frac{u^{2}}{2} d u = \int 4 d x$

Step 2.2.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int u^{2} d u = \int 4 d x$

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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Step 2.2.5.1

Rewrite $\frac{1}{2} (\frac{1}{3} u^{3} + C_{1})$ as $\frac{1}{2} \cdot \frac{1}{3} u^{3} + C_{1}$ .

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

Step 2.2.5.2

Simplify.

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Step 2.2.5.2.1

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

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

Step 2.2.5.2.2

Multiply $2$ by $3$ .

$\frac{1}{6} u^{3} + C_{1} = \int 4 d x$

Step 2.2.6

Replace all occurrences of $u$ with $2 y + 1$ .

$\frac{1}{6} {(2 y + 1)}^{3} + C_{1} = \int 4 d x$

Step 2.3

Apply the constant rule.

$\frac{1}{6} {(2 y + 1)}^{3} + C_{1} = 4 x + C_{2}$

Step 2.4

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

$\frac{1}{6} {(2 y + 1)}^{3} = 4 x + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $6$ .

$6 (\frac{1}{6} {(2 y + 1)}^{3}) = 6 (4 x + K)$

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 $6 (\frac{1}{6} {(2 y + 1)}^{3})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{6}$ and ${(2 y + 1)}^{3}$ .

$6 \frac{{(2 y + 1)}^{3}}{6} = 6 (4 x + K)$

Step 3.2.1.1.2

Cancel the common factor of $6$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$6 \frac{{(2 y + 1)}^{3}}{6} = 6 (4 x + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 y + 1)}^{3} = 6 (4 x + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $6 (4 x + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

${(2 y + 1)}^{3} = 6 (4 x) + 6 K$

Step 3.2.2.1.2

Multiply $4$ by $6$ .

${(2 y + 1)}^{3} = 24 x + 6 K$

Step 3.3

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

$2 y + 1 = \sqrt[3]{24 x + 6 K}$

Step 3.4

Simplify $\sqrt[3]{24 x + 6 K}$ .

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Step 3.4.1

Rewrite.

$2 y + 1 = 0 + 0 + \sqrt[3]{24 x + 6 K}$

Step 3.4.2

Simplify by adding zeros.

$2 y + 1 = \sqrt[3]{24 x + 6 K}$

Step 3.4.3

Factor $6$ out of $24 x + 6 K$ .

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Step 3.4.3.1

Factor $6$ out of $24 x$ .

$2 y + 1 = \sqrt[3]{6 (4 x) + 6 K}$

Step 3.4.3.2

Factor $6$ out of $6 K$ .

$2 y + 1 = \sqrt[3]{6 (4 x) + 6 K}$

Step 3.4.3.3

Factor $6$ out of $6 (4 x) + 6 K$ .

$2 y + 1 = \sqrt[3]{6 (4 x + K)}$

Step 3.5

Subtract $1$ from both sides of the equation.

$2 y = \sqrt[3]{6 (4 x + K)} - 1$

Step 3.6

Divide each term in $2 y = \sqrt[3]{6 (4 x + K)} - 1$ by $2$ and simplify.

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Step 3.6.1

Divide each term in $2 y = \sqrt[3]{6 (4 x + K)} - 1$ by $2$ .

$\frac{2 y}{2} = \frac{\sqrt[3]{6 (4 x + K)}}{2} + \frac{- 1}{2}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\sqrt[3]{6 (4 x + K)}}{2} + \frac{- 1}{2}$

Step 3.6.2.1.2

Divide $y$ by $1$ .

$y = \frac{\sqrt[3]{6 (4 x + K)}}{2} + \frac{- 1}{2}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

$y = \frac{\sqrt[3]{6 (4 x + K)}}{2} - \frac{1}{2}$

Step 3.6.3.2

Combine the numerators over the common denominator.

$y = \frac{\sqrt[3]{6 (4 x + K)} - 1}{2}$