Calculus Examples

$\frac{2 d y}{3 d x} = {(x - 2)}^{2}$

Step 1

Separate the variables.

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Step 1.1

Factor $\frac{d y}{d x}$ out.

$\frac{2}{3} \frac{d y}{d x} = {(x - 2)}^{2}$

Step 1.2

Rewrite the equation.

$\frac{2}{3} d y = {(x - 2)}^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = x - 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Differentiate $x - 2$ .

$\frac{d}{d x} [x - 2]$

Step 2.3.1.1.2

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

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

Step 2.3.1.1.3

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

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

Step 2.3.1.1.4

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

$1 + 0$

Step 2.3.1.1.5

Add $1$ and $0$ .

$1$

Step 2.3.1.2

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

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u$ with $x - 2$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $\frac{3}{2}$ .

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

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 $\frac{3}{2} (\frac{2}{3} y)$ .

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Step 3.2.1.1.1

Combine $\frac{2}{3}$ and $y$ .

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.1.3

Cancel the common factor of $2$ .

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Step 3.2.1.1.3.1

Factor $2$ out of $2 y$ .

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

Step 3.2.1.1.3.2

Cancel the common factor.

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

Step 3.2.1.1.3.3

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{3}{2} (\frac{1}{3} {(x - 2)}^{3} + K)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{3}$ and ${(x - 2)}^{3}$ .

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

Step 3.2.2.1.2

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

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

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

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

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

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

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

Step 3.2.2.1.3.3

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.3.1

Cancel the common factor.

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

Step 3.2.2.1.3.3.2

Rewrite the expression.

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

Step 3.2.2.1.4

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

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

Step 3.2.2.1.5

Simplify terms.

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Step 3.2.2.1.5.1

Apply the distributive property.

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

Step 3.2.2.1.5.2

Combine $\frac{1}{2}$ and ${(x - 2)}^{3}$ .

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

Step 3.2.2.1.6

Multiply $\frac{1}{2} (3 K)$ .

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Step 3.2.2.1.6.1

Combine $3$ and $\frac{1}{2}$ .

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

Step 3.2.2.1.6.2

Combine $\frac{3}{2}$ and $K$ .

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

Step 3.2.2.1.7

Combine the numerators over the common denominator.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Use the Binomial Theorem.

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

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Multiply $- 2$ by $3$ .

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

Step 3.3.2.2

Raise $- 2$ to the power of $2$ .

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

Step 3.3.2.3

Multiply $4$ by $3$ .

$y = \frac{x^{3} - 6 x^{2} + 12 x + {(- 2)}^{3} + 3 K}{2}$

Step 3.3.2.4

Raise $- 2$ to the power of $3$ .

$y = \frac{x^{3} - 6 x^{2} + 12 x - 8 + 3 K}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{x^{3} - 6 x^{2} + 12 x + K}{2}$