Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{a}{{(1 + \frac{x}{b})}^{2}} d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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Step 2.3.2.1

Let $u = 1 + \frac{x}{b}$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $1 + \frac{x}{b}$ .

$\frac{d}{d x} [1 + \frac{x}{b}]$

Step 2.3.2.1.2

Differentiate.

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Step 2.3.2.1.2.1

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

$\frac{d}{d x} [1] + \frac{d}{d x} [\frac{x}{b}]$

Step 2.3.2.1.2.2

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

$0 + \frac{d}{d x} [\frac{x}{b}]$

Step 2.3.2.1.3

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

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Step 2.3.2.1.3.1

Since $\frac{1}{b}$ is constant with respect to $x$ , the derivative of $\frac{x}{b}$ with respect to $x$ is $\frac{1}{b} \frac{d}{d x} [x]$ .

$0 + \frac{1}{b} \frac{d}{d x} [x]$

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 $n = 1$ .

$0 + \frac{1}{b} \cdot 1$

Step 2.3.2.1.3.3

Multiply $\frac{1}{b}$ by $1$ .

$0 + \frac{1}{b}$

Step 2.3.2.1.4

Add $0$ and $\frac{1}{b}$ .

$\frac{1}{b}$

Step 2.3.2.2

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

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{b}$ .

$\frac{1}{2} y^{2} + C_{1} = a \int \frac{1}{u^{2}} (1 b) d u$

Step 2.3.3.2

Multiply $b$ by $1$ .

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

Step 2.3.3.3

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

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

Step 2.3.4

Since $b$ is constant with respect to $u$ , move $b$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = a (b \int \frac{1}{u^{2}} d u)$

Step 2.3.5

Apply basic rules of exponents.

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Step 2.3.5.1

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

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

Step 2.3.5.2

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

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{2} y^{2} + C_{1} = a b \int u^{- 2} d u$

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Combine $a$ and $\frac{1}{u}$ .

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

Step 2.3.7.2.2

Combine $b$ and $\frac{a}{u}$ .

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

Step 2.3.8

Replace all occurrences of $u$ with $1 + \frac{x}{b}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{b a}{1 + \frac{x}{b}} + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = - \frac{b a}{1 + \frac{x}{b}} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- \frac{b a}{1 + \frac{x}{b}} + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Simplify the denominator.

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Step 3.2.2.1.1.1.1

Write $1$ as a fraction with a common denominator.

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

Step 3.2.2.1.1.1.2

Combine the numerators over the common denominator.

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

Step 3.2.2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2.2.1.1.3

Multiply $b a \frac{b}{b + x}$ .

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Step 3.2.2.1.1.3.1

Combine $b$ and $\frac{b}{b + x}$ .

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

Step 3.2.2.1.1.3.2

Raise $b$ to the power of $1$ .

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

Step 3.2.2.1.1.3.3

Raise $b$ to the power of $1$ .

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

Step 3.2.2.1.1.3.4

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

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

Step 3.2.2.1.1.3.5

Add $1$ and $1$ .

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

Step 3.2.2.1.1.3.6

Combine $a$ and $\frac{b^{2}}{b + x}$ .

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

Step 3.2.2.1.2

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

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

Step 3.2.2.1.3

Combine the numerators over the common denominator.

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

Step 3.2.2.1.4

Apply the distributive property.

$y^{2} = 2 \frac{- a b^{2} + K b + K x}{b + x}$

Step 3.2.2.1.5

Simplify terms.

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Step 3.2.2.1.5.1

Combine $2$ and $\frac{- a b^{2} + K b + K x}{b + x}$ .

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

Step 3.2.2.1.5.2

Factor $- 1$ out of $- a b^{2}$ .

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

Step 3.2.2.1.5.3

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

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

Step 3.2.2.1.5.4

Factor $- 1$ out of $- (a b^{2}) - (- K b)$ .

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

Step 3.2.2.1.5.5

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

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

Step 3.2.2.1.5.6

Factor $- 1$ out of $- (a b^{2} - K b) - (- K x)$ .

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

Step 3.2.2.1.5.7

Simplify the expression.

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Step 3.2.2.1.5.7.1

Rewrite $- (a b^{2} - K b - K x)$ as $- 1 (a b^{2} - K b - K x)$ .

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

Step 3.2.2.1.5.7.2

Move the negative in front of the fraction.

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

Step 3.3

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

$y = \pm \sqrt{- \frac{2 (a b^{2} - K b - K x)}{b + x}}$

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.

$y = \sqrt{- \frac{2 (a b^{2} - K b - K x)}{b + x}}$

Step 3.4.2

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

$y = - \sqrt{- \frac{2 (a b^{2} - K b - K x)}{b + x}}$

Step 3.4.3

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

$y = \sqrt{- \frac{2 (a b^{2} - K b - K x)}{b + x}}$

$y = - \sqrt{- \frac{2 (a b^{2} - K b - K x)}{b + x}}$

$y = \sqrt{- \frac{2 (a b^{2} - K b - K x)}{b + x}}$

$y = - \sqrt{- \frac{2 (a b^{2} - K b - K x)}{b + x}}$

$y = \sqrt{- \frac{2 (a b^{2} - K b - K x)}{b + x}}$

$y = - \sqrt{- \frac{2 (a b^{2} - K b - K x)}{b + x}}$

Step 4

Simplify the constant of integration.

$y = \sqrt{- \frac{2 (a b^{2} + K b + K x)}{b + x}}$

$y = - \sqrt{- \frac{2 (a b^{2} + K b + K x)}{b + x}}$