Calculus Examples

$\frac{d y}{d x} = \frac{a x + b}{c x + d}$

Step 1

Rewrite the equation.

$d y = \frac{a x + b}{c x + d} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{a x + b}{c x + d} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{a x + b}{c x + d} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Differentiate $c x + d$ .

$\frac{d}{d x} [c x + d]$

Step 2.3.1.1.2

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

$\frac{d}{d x} [c x] + \frac{d}{d x} [d]$

Step 2.3.1.1.3

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

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Step 2.3.1.1.3.1

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

$c \frac{d}{d x} [x] + \frac{d}{d x} [d]$

Step 2.3.1.1.3.2

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

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

Step 2.3.1.1.3.3

Multiply $c$ by $1$ .

$c + \frac{d}{d x} [d]$

Step 2.3.1.1.4

Differentiate using the Constant Rule.

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Step 2.3.1.1.4.1

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

$c + 0$

Step 2.3.1.1.4.2

Add $c$ and $0$ .

$c$

Step 2.3.1.2

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

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

Step 2.3.2

Multiply $\frac{a (\frac{u}{c} - \frac{d}{c}) + b}{u}$ by $\frac{1}{c}$ .

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

Step 2.3.3

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

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

Step 2.3.4

Split the fraction into multiple fractions.

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

Step 2.3.5

Split the single integral into multiple integrals.

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

Multiply $\frac{\frac{u}{c} - \frac{d}{c}}{u}$ by $\frac{c}{c}$ .

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

Step 2.3.7.2

Combine.

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

Step 2.3.7.3

Apply the distributive property.

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

Step 2.3.7.4

Cancel the common factor of $c$ .

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Step 2.3.7.4.1

Cancel the common factor.

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

Step 2.3.7.4.2

Rewrite the expression.

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

Step 2.3.7.5

Combine $c$ and $\frac{d}{c}$ .

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

Step 2.3.7.6

Cancel the common factor of $c$ .

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Step 2.3.7.6.1

Cancel the common factor.

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

Step 2.3.7.6.2

Divide $d$ by $1$ .

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

Step 2.3.8

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

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

Step 2.3.9

Split the fraction into multiple fractions.

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

Step 2.3.10

Split the single integral into multiple integrals.

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

Step 2.3.11

Cancel the common factor of $u$ .

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Step 2.3.11.1

Cancel the common factor.

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

Step 2.3.11.2

Rewrite the expression.

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

Step 2.3.12

Apply the constant rule.

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

Step 2.3.13

Move the negative in front of the fraction.

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

Step 2.3.14

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 2.3.15

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

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

Step 2.3.16

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$y + C_{1} = \frac{1}{c} (a (\frac{1}{c} (u + C_{2} - (d (\ln (| u |) + C_{3})))) + \int \frac{b}{u} d u)$

Step 2.3.17

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

$y + C_{1} = \frac{1}{c} (\frac{a}{c} (u + C_{2} - d (\ln (| u |) + C_{3})) + \int \frac{b}{u} d u)$

Step 2.3.18

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

$y + C_{1} = \frac{1}{c} (\frac{a}{c} (u + C_{2} - d (\ln (| u |) + C_{3})) + b \int \frac{1}{u} d u)$

Step 2.3.19

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$y + C_{1} = \frac{1}{c} (\frac{a}{c} (u + C_{2} - d (\ln (| u |) + C_{3})) + b (\ln (| u |) + C_{4}))$

Step 2.3.20

Simplify.

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Step 2.3.20.1

Simplify.

$y + C_{1} = \frac{1}{c} (\frac{a (u - d \ln (| u |))}{c} + b \ln (| u |)) + C_{5}$

Step 2.3.20.2

Simplify.

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Step 2.3.20.2.1

To write $b \ln (| u |)$ as a fraction with a common denominator, multiply by $\frac{c}{c}$ .

$y + C_{1} = \frac{1}{c} (\frac{a (u - d \ln (| u |))}{c} + \frac{b \ln (| u |) c}{c}) + C_{5}$

Step 2.3.20.2.2

Combine the numerators over the common denominator.

$y + C_{1} = \frac{1}{c} \cdot \frac{a (u - d \ln (| u |)) + b \ln (| u |) c}{c} + C_{5}$

Step 2.3.20.2.3

Multiply $\frac{1}{c}$ by $\frac{a (u - d \ln (| u |)) + b \ln (| u |) c}{c}$ .

$y + C_{1} = \frac{a (u - d \ln (| u |)) + b \ln (| u |) c}{c \cdot c} + C_{5}$

Step 2.3.20.2.4

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

$y + C_{1} = \frac{a (u - d \ln (| u |)) + b \ln (| u |) c}{c^{1} c} + C_{5}$

Step 2.3.20.2.5

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

$y + C_{1} = \frac{a (u - d \ln (| u |)) + b \ln (| u |) c}{c^{1} c^{1}} + C_{5}$

Step 2.3.20.2.6

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

$y + C_{1} = \frac{a (u - d \ln (| u |)) + b \ln (| u |) c}{c^{1 + 1}} + C_{5}$

Step 2.3.20.2.7

Add $1$ and $1$ .

$y + C_{1} = \frac{a (u - d \ln (| u |)) + b \ln (| u |) c}{c^{2}} + C_{5}$

Step 2.3.21

Replace all occurrences of $u$ with $c x + d$ .

$y + C_{1} = \frac{a (c x + d - d \ln (| c x + d |)) + b \ln (| c x + d |) c}{c^{2}} + C_{5}$

Step 2.4

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

$y = \frac{a (c x + d - d \ln (| c x + d |)) + b \ln (| c x + d |) c}{c^{2}} + C$