Calculus Examples

$\frac{d y}{d x} = \frac{3 + y}{1 - 2 x}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{3 + y}$ .

$\frac{1}{3 + y} \frac{d y}{d x} = \frac{1}{3 + y} \cdot \frac{3 + y}{1 - 2 x}$

Step 1.2

Cancel the common factor of $3 + y$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{3 + y} \frac{d y}{d x} = \frac{1}{3 + y} \cdot \frac{3 + y}{1 - 2 x}$

Step 1.2.2

Rewrite the expression.

$\frac{1}{3 + y} \frac{d y}{d x} = \frac{1}{1 - 2 x}$

Step 1.3

Rewrite the equation.

$\frac{1}{3 + y} d y = \frac{1}{1 - 2 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{3 + y} d y = \int \frac{1}{1 - 2 x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u_{1} = 3 + y$ . Then $d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.1.1

Let $u_{1} = 3 + y$ . Find $\frac{d u_{1}}{d y}$ .

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Step 2.2.1.1.1

Differentiate $3 + y$ .

$\frac{d}{d y} [3 + y]$

Step 2.2.1.1.2

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

$\frac{d}{d y} [3] + \frac{d}{d y} [y]$

Step 2.2.1.1.3

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [y]$

Step 2.2.1.1.4

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

$0 + 1$

Step 2.2.1.1.5

Add $0$ and $1$ .

$1$

Step 2.2.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{u_{1}} d u_{1} = \int \frac{1}{1 - 2 x} d x$

Step 2.2.2

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

$\ln (| u_{1} |) + C_{1} = \int \frac{1}{1 - 2 x} d x$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $3 + y$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u_{2} = 1 - 2 x$ . Then $d u_{2} = - 2 d x$ , so $- \frac{1}{2} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.1.1

Let $u_{2} = 1 - 2 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.1.1.1

Differentiate $1 - 2 x$ .

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

Step 2.3.1.1.2

Differentiate.

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Step 2.3.1.1.2.1

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

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

Step 2.3.1.1.2.2

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

$0 + \frac{d}{d x} [- 2 x]$

Step 2.3.1.1.3

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

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Step 2.3.1.1.3.1

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

$0 - 2 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 2 \cdot 1$

Step 2.3.1.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 2.3.1.1.4

Subtract $2$ from $0$ .

$- 2$

Step 2.3.1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\ln (| 3 + y |) + C_{1} = \int \frac{1}{u_{2}} \cdot \frac{1}{- 2} d u_{2}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Move the negative in front of the fraction.

$\ln (| 3 + y |) + C_{1} = \int \frac{1}{u_{2}} (- \frac{1}{2}) d u_{2}$

Step 2.3.2.2

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

$\ln (| 3 + y |) + C_{1} = \int - \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 2.3.2.3

Move $2$ to the left of $u_{2}$ .

$\ln (| 3 + y |) + C_{1} = \int - \frac{1}{2 u_{2}} d u_{2}$

Step 2.3.3

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$\ln (| 3 + y |) + C_{1} = - \int \frac{1}{2 u_{2}} d u_{2}$

Step 2.3.4

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

$\ln (| 3 + y |) + C_{1} = - (\frac{1}{2} \int \frac{1}{u_{2}} d u_{2})$

Step 2.3.5

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\ln (| 3 + y |) + C_{1} = - \frac{1}{2} (\ln (| u_{2} |) + C_{2})$

Step 2.3.6

Simplify.

$\ln (| 3 + y |) + C_{1} = - \frac{1}{2} \ln (| u_{2} |) + C_{2}$

Step 2.3.7

Replace all occurrences of $u_{2}$ with $1 - 2 x$ .

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

Step 2.4

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

$\ln (| 3 + y |) = - \frac{1}{2} \ln (| 1 - 2 x |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Simplify the right side.

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Step 3.1.1

Combine $\ln (| 1 - 2 x |)$ and $\frac{1}{2}$ .

$\ln (| 3 + y |) = - \frac{\ln (| 1 - 2 x |)}{2} + C$

Step 3.2

Move all the terms containing a logarithm to the left side of the equation.

$\ln (| 3 + y |) + \frac{\ln (| 1 - 2 x |)}{2} = C$

Step 3.3

To write $\ln (| 3 + y |)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\ln (| 3 + y |) \cdot \frac{2}{2} + \frac{\ln (| 1 - 2 x |)}{2} = C$

Step 3.4

Simplify terms.

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Step 3.4.1

Combine $\ln (| 3 + y |)$ and $\frac{2}{2}$ .

$\frac{\ln (| 3 + y |) \cdot 2}{2} + \frac{\ln (| 1 - 2 x |)}{2} = C$

Step 3.4.2

Combine the numerators over the common denominator.

$\frac{\ln (| 3 + y |) \cdot 2 + \ln (| 1 - 2 x |)}{2} = C$

Step 3.5

Move $2$ to the left of $\ln (| 3 + y |)$ .

$\frac{2 \ln (| 3 + y |) + \ln (| 1 - 2 x |)}{2} = C$

Step 3.6

Simplify the left side.

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Step 3.6.1

Simplify $\frac{2 \ln (| 3 + y |) + \ln (| 1 - 2 x |)}{2}$ .

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Step 3.6.1.1

Simplify the numerator.

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Step 3.6.1.1.1

Simplify $2 \ln (| 3 + y |)$ by moving $2$ inside the logarithm.

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

Step 3.6.1.1.2

Remove the absolute value in ${| 3 + y |}^{2}$ because exponentiations with even powers are always positive.

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

Step 3.6.1.1.3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 3.6.1.2

Rewrite $\frac{\ln ({(3 + y)}^{2} | 1 - 2 x |)}{2}$ as $\frac{1}{2} \ln ({(3 + y)}^{2} | 1 - 2 x |)$ .

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

Step 3.6.1.3

Simplify $\frac{1}{2} \ln ({(3 + y)}^{2} | 1 - 2 x |)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 3.6.1.4

Apply the product rule to ${(3 + y)}^{2} | 1 - 2 x |$ .

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

Step 3.6.1.5

Multiply the exponents in ${({(3 + y)}^{2})}^{\frac{1}{2}}$ .

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Step 3.6.1.5.1

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

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

Step 3.6.1.5.2

Cancel the common factor of $2$ .

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Step 3.6.1.5.2.1

Cancel the common factor.

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

Step 3.6.1.5.2.2

Rewrite the expression.

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

Step 3.6.1.6

Simplify.

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

Step 3.6.1.7

Apply the distributive property.

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

Step 3.7

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (3 {| 1 - 2 x |}^{\frac{1}{2}} + y {| 1 - 2 x |}^{\frac{1}{2}})} = e^{C}$

Step 3.8

Rewrite $\ln (3 {| 1 - 2 x |}^{\frac{1}{2}} + y {| 1 - 2 x |}^{\frac{1}{2}}) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = 3 {| 1 - 2 x |}^{\frac{1}{2}} + y {| 1 - 2 x |}^{\frac{1}{2}}$

Step 3.9

Solve for $y$ .

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Step 3.9.1

Rewrite the equation as $3 {| 1 - 2 x |}^{\frac{1}{2}} + y {| 1 - 2 x |}^{\frac{1}{2}} = e^{C}$ .

$3 {| 1 - 2 x |}^{\frac{1}{2}} + y {| 1 - 2 x |}^{\frac{1}{2}} = e^{C}$

Step 3.9.2

Subtract $3 {| 1 - 2 x |}^{\frac{1}{2}}$ from both sides of the equation.

$y {| 1 - 2 x |}^{\frac{1}{2}} = e^{C} - 3 {| 1 - 2 x |}^{\frac{1}{2}}$

Step 3.9.3

Divide each term in $y {| 1 - 2 x |}^{\frac{1}{2}} = e^{C} - 3 {| 1 - 2 x |}^{\frac{1}{2}}$ by ${| 1 - 2 x |}^{\frac{1}{2}}$ and simplify.

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Step 3.9.3.1

Divide each term in $y {| 1 - 2 x |}^{\frac{1}{2}} = e^{C} - 3 {| 1 - 2 x |}^{\frac{1}{2}}$ by ${| 1 - 2 x |}^{\frac{1}{2}}$ .

$\frac{y {| 1 - 2 x |}^{\frac{1}{2}}}{{| 1 - 2 x |}^{\frac{1}{2}}} = \frac{e^{C}}{{| 1 - 2 x |}^{\frac{1}{2}}} + \frac{- 3 {| 1 - 2 x |}^{\frac{1}{2}}}{{| 1 - 2 x |}^{\frac{1}{2}}}$

Step 3.9.3.2

Simplify the left side.

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Step 3.9.3.2.1

Cancel the common factor.

$\frac{y {| 1 - 2 x |}^{\frac{1}{2}}}{{| 1 - 2 x |}^{\frac{1}{2}}} = \frac{e^{C}}{{| 1 - 2 x |}^{\frac{1}{2}}} + \frac{- 3 {| 1 - 2 x |}^{\frac{1}{2}}}{{| 1 - 2 x |}^{\frac{1}{2}}}$

Step 3.9.3.2.2

Divide $y$ by $1$ .

$y = \frac{e^{C}}{{| 1 - 2 x |}^{\frac{1}{2}}} + \frac{- 3 {| 1 - 2 x |}^{\frac{1}{2}}}{{| 1 - 2 x |}^{\frac{1}{2}}}$

Step 3.9.3.3

Simplify the right side.

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Step 3.9.3.3.1

Combine the numerators over the common denominator.

$y = \frac{e^{C} - 3 {| 1 - 2 x |}^{\frac{1}{2}}}{{| 1 - 2 x |}^{\frac{1}{2}}}$

Step 4

Simplify the constant of integration.

$y = \frac{C - 3 {| 1 - 2 x |}^{\frac{1}{2}}}{{| 1 - 2 x |}^{\frac{1}{2}}}$