Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{{(1 - 3 y)}^{2}}$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Combine.

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

Step 1.2.2

Cancel the common factor of ${(1 - 3 y)}^{2}$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 1 - 3 y$ . Then $d u = - 3 d y$ , so $- \frac{1}{3} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = 1 - 3 y$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $1 - 3 y$ .

$\frac{d}{d y} [1 - 3 y]$

Step 2.2.1.1.2

Differentiate.

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Step 2.2.1.1.2.1

By the Sum Rule, the derivative of $1 - 3 y$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [- 3 y]$ .

$\frac{d}{d y} [1] + \frac{d}{d y} [- 3 y]$

Step 2.2.1.1.2.2

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

$0 + \frac{d}{d y} [- 3 y]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d y} [- 3 y]$ .

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Step 2.2.1.1.3.1

Since $- 3$ is constant with respect to $y$ , the derivative of $- 3 y$ with respect to $y$ is $- 3 \frac{d}{d y} [y]$ .

$0 - 3 \frac{d}{d y} [y]$

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 $n = 1$ .

$0 - 3 \cdot 1$

Step 2.2.1.1.3.3

Multiply $- 3$ by $1$ .

$0 - 3$

Step 2.2.1.1.4

Subtract $3$ from $0$ .

$- 3$

Step 2.2.1.2

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

$\int \frac{1}{u^{2}} \cdot \frac{1}{- 3} d u = \int \frac{1}{x} d x$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move the negative in front of the fraction.

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

Step 2.2.2.2

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

$\int - \frac{1}{u^{2} \cdot 3} d u = \int \frac{1}{x} d x$

Step 2.2.2.3

Move $3$ to the left of $u^{2}$ .

$\int - \frac{1}{3 u^{2}} d u = \int \frac{1}{x} d x$

Step 2.2.3

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

$- \int \frac{1}{3 u^{2}} d u = \int \frac{1}{x} d x$

Step 2.2.4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

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

Step 2.2.5

Apply basic rules of exponents.

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Step 2.2.5.1

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

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

Step 2.2.5.2

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

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Step 2.2.5.2.1

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

$- \frac{1}{3} \int u^{2 \cdot - 1} d u = \int \frac{1}{x} d x$

Step 2.2.5.2.2

Multiply $2$ by $- 1$ .

$- \frac{1}{3} \int u^{- 2} d u = \int \frac{1}{x} d x$

Step 2.2.6

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

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

Step 2.2.7

Simplify.

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Step 2.2.7.1

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

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

Step 2.2.7.2

Simplify.

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Step 2.2.7.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.7.2.2

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

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

Step 2.2.7.2.3

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

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

Step 2.2.8

Replace all occurrences of $u$ with $1 - 3 y$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 (1 - 3 y), 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$3 (1 - 3 y)$

Step 3.2

Multiply each term in $\frac{1}{3 (1 - 3 y)} = \ln (| x |) + C$ by $3 (1 - 3 y)$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{1}{3 (1 - 3 y)} = \ln (| x |) + C$ by $3 (1 - 3 y)$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2.2

Cancel the common factor of $3$ .

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Step 3.2.2.2.1

Cancel the common factor.

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

Step 3.2.2.2.2

Rewrite the expression.

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

Step 3.2.2.3

Cancel the common factor of $1 - 3 y$ .

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Step 3.2.2.3.1

Cancel the common factor.

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

Step 3.2.2.3.2

Rewrite the expression.

$1 = \ln (| x |) (3 (1 - 3 y)) + C (3 (1 - 3 y))$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Rewrite using the commutative property of multiplication.

$1 = 3 \ln (| x |) (1 - 3 y) + C (3 (1 - 3 y))$

Step 3.2.3.1.2

Simplify $3 \ln (| x |)$ by moving $3$ inside the logarithm.

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

Step 3.2.3.1.3

Apply the distributive property.

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

Step 3.2.3.1.4

Multiply $\ln ({| x |}^{3})$ by $1$ .

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

Step 3.2.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 3.2.3.1.6

Simplify each term.

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Step 3.2.3.1.6.1

Simplify $- 3 \ln ({| x |}^{3})$ by moving $3$ inside the logarithm.

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

Step 3.2.3.1.6.2

Multiply the exponents in ${({| x |}^{3})}^{3}$ .

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Step 3.2.3.1.6.2.1

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

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

Step 3.2.3.1.6.2.2

Multiply $3$ by $3$ .

$1 = \ln ({| x |}^{3}) - \ln ({| x |}^{9}) y + C (3 (1 - 3 y))$

Step 3.2.3.1.7

Rewrite using the commutative property of multiplication.

$1 = \ln ({| x |}^{3}) - \ln ({| x |}^{9}) y + 3 C (1 - 3 y)$

Step 3.2.3.1.8

Apply the distributive property.

$1 = \ln ({| x |}^{3}) - \ln ({| x |}^{9}) y + 3 C \cdot 1 + 3 C (- 3 y)$

Step 3.2.3.1.9

Multiply $3$ by $1$ .

$1 = \ln ({| x |}^{3}) - \ln ({| x |}^{9}) y + 3 C + 3 C (- 3 y)$

Step 3.2.3.1.10

Rewrite using the commutative property of multiplication.

$1 = \ln ({| x |}^{3}) - \ln ({| x |}^{9}) y + 3 C + 3 \cdot - 3 C y$

Step 3.2.3.1.11

Multiply $3$ by $- 3$ .

$1 = \ln ({| x |}^{3}) - \ln ({| x |}^{9}) y + 3 C - 9 C y$

Step 3.2.3.2

Reorder factors in $\ln ({| x |}^{3}) - \ln ({| x |}^{9}) y + 3 C - 9 C y$ .

$1 = \ln ({| x |}^{3}) - y \ln ({| x |}^{9}) + 3 C - 9 C y$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $\ln ({| x |}^{3}) - y \ln ({| x |}^{9}) + 3 C - 9 C y = 1$ .

$\ln ({| x |}^{3}) - y \ln ({| x |}^{9}) + 3 C - 9 C y = 1$

Step 3.3.2

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

$\ln ({| x |}^{3}) - y \ln ({| x |}^{9}) = - 3 C + 9 C y + 1$

Step 3.3.3

Subtract $9 C y$ from both sides of the equation.

$\ln ({| x |}^{3}) - y \ln ({| x |}^{9}) - 9 C y = - 3 C + 1$

Step 3.3.4

Subtract $\ln ({| x |}^{3})$ from both sides of the equation.

$- y \ln ({| x |}^{9}) - 9 C y = - 3 C + 1 - \ln ({| x |}^{3})$

Step 3.3.5

Factor $y$ out of $- y \ln ({| x |}^{9}) - 9 C y$ .

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Step 3.3.5.1

Factor $y$ out of $- y \ln ({| x |}^{9})$ .

$y (- 1 \ln ({| x |}^{9})) - 9 C y = - 3 C + 1 - \ln ({| x |}^{3})$

Step 3.3.5.2

Factor $y$ out of $- 9 C y$ .

$y (- 1 \ln ({| x |}^{9})) + y (- 9 C) = - 3 C + 1 - \ln ({| x |}^{3})$

Step 3.3.5.3

Factor $y$ out of $y (- 1 \ln ({| x |}^{9})) + y (- 9 C)$ .

$y (- 1 \ln ({| x |}^{9}) - 9 C) = - 3 C + 1 - \ln ({| x |}^{3})$

Step 3.3.6

Rewrite $- 1 \ln ({| x |}^{9})$ as $- \ln ({| x |}^{9})$ .

$y (- \ln ({| x |}^{9}) - 9 C) = - 3 C + 1 - \ln ({| x |}^{3})$

Step 3.3.7

Divide each term in $y (- \ln ({| x |}^{9}) - 9 C) = - 3 C + 1 - \ln ({| x |}^{3})$ by $- \ln ({| x |}^{9}) - 9 C$ and simplify.

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Step 3.3.7.1

Divide each term in $y (- \ln ({| x |}^{9}) - 9 C) = - 3 C + 1 - \ln ({| x |}^{3})$ by $- \ln ({| x |}^{9}) - 9 C$ .

$\frac{y (- \ln ({| x |}^{9}) - 9 C)}{- \ln ({| x |}^{9}) - 9 C} = \frac{- 3 C}{- \ln ({| x |}^{9}) - 9 C} + \frac{1}{- \ln ({| x |}^{9}) - 9 C} + \frac{- \ln ({| x |}^{3})}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.2

Simplify the left side.

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Step 3.3.7.2.1

Cancel the common factor of $- \ln ({| x |}^{9}) - 9 C$ .

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Step 3.3.7.2.1.1

Cancel the common factor.

Step 3.3.7.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 C}{- \ln ({| x |}^{9}) - 9 C} + \frac{1}{- \ln ({| x |}^{9}) - 9 C} + \frac{- \ln ({| x |}^{3})}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3

Simplify the right side.

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Step 3.3.7.3.1

Simplify each term.

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Step 3.3.7.3.1.1

Move the negative in front of the fraction.

$y = - \frac{3 C}{- \ln ({| x |}^{9}) - 9 C} + \frac{1}{- \ln ({| x |}^{9}) - 9 C} + \frac{- \ln ({| x |}^{3})}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3.1.2

Move the negative in front of the fraction.

$y = - \frac{3 C}{- \ln ({| x |}^{9}) - 9 C} + \frac{1}{- \ln ({| x |}^{9}) - 9 C} - \frac{\ln ({| x |}^{3})}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3.2

Simplify terms.

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Step 3.3.7.3.2.1

Combine the numerators over the common denominator.

$y = \frac{- 3 C + 1}{- \ln ({| x |}^{9}) - 9 C} - \frac{\ln ({| x |}^{3})}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3.2.2

Combine the numerators over the common denominator.

$y = \frac{- 3 C + 1 - \ln ({| x |}^{3})}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3.2.3

Factor $- 1$ out of $- 3 C$ .

$y = \frac{- (3 C) + 1 - \ln ({| x |}^{3})}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3.2.4

Rewrite $1$ as $- 1 (- 1)$ .

$y = \frac{- (3 C) - 1 (- 1) - \ln ({| x |}^{3})}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3.2.5

Factor $- 1$ out of $- (3 C) - 1 (- 1)$ .

$y = \frac{- (3 C - 1) - \ln ({| x |}^{3})}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3.2.6

Factor $- 1$ out of $- (3 C - 1) - \ln ({| x |}^{3})$ .

$y = \frac{- (3 C - 1 + \ln ({| x |}^{3}))}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3.2.7

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

$y = \frac{- 1 (3 C - 1 + \ln ({| x |}^{3}))}{- \ln ({| x |}^{9}) - 9 C}$

Step 3.3.7.3.2.8

Factor $- 1$ out of $- 9 C$ .

$y = \frac{- 1 (3 C - 1 + \ln ({| x |}^{3}))}{- \ln ({| x |}^{9}) - (9 C)}$

Step 3.3.7.3.2.9

Factor $- 1$ out of $- \ln ({| x |}^{9}) - (9 C)$ .

$y = \frac{- 1 (3 C - 1 + \ln ({| x |}^{3}))}{- (\ln ({| x |}^{9}) + 9 C)}$

Step 3.3.7.3.2.10

Rewrite $- (\ln ({| x |}^{9}) + 9 C)$ as $- 1 (\ln ({| x |}^{9}) + 9 C)$ .

$y = \frac{- 1 (3 C - 1 + \ln ({| x |}^{3}))}{- 1 (\ln ({| x |}^{9}) + 9 C)}$

Step 3.3.7.3.2.11

Cancel the common factor.

$y = \frac{- 1 (3 C - 1 + \ln ({| x |}^{3}))}{- 1 (\ln ({| x |}^{9}) + 9 C)}$

Step 3.3.7.3.2.12

Rewrite the expression.

$y = \frac{3 C - 1 + \ln ({| x |}^{3})}{\ln ({| x |}^{9}) + 9 C}$

Step 4

Simplify the constant of integration.

$y = \frac{C + \ln ({| x |}^{3})}{\ln ({| x |}^{9}) + C}$