Calculus Examples

$z + u (\frac{d z}{d u}) = \frac{z}{1 - z}$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d z}{d u}$ .

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Step 1.1.1

Subtract $z$ from both sides of the equation.

$u \frac{d z}{d u} = \frac{z}{1 - z} - z$

Step 1.1.2

Divide each term in $u \frac{d z}{d u} = \frac{z}{1 - z} - z$ by $u$ and simplify.

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Step 1.1.2.1

Divide each term in $u \frac{d z}{d u} = \frac{z}{1 - z} - z$ by $u$ .

$\frac{u \frac{d z}{d u}}{u} = \frac{\frac{z}{1 - z}}{u} + \frac{- z}{u}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $u$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{u \frac{d z}{d u}}{u} = \frac{\frac{z}{1 - z}}{u} + \frac{- z}{u}$

Step 1.1.2.2.1.2

Divide $\frac{d z}{d u}$ by $1$ .

$\frac{d z}{d u} = \frac{\frac{z}{1 - z}}{u} + \frac{- z}{u}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Combine the numerators over the common denominator.

$\frac{d z}{d u} = \frac{\frac{z}{1 - z} - z}{u}$

Step 1.1.2.3.2

To write $- z$ as a fraction with a common denominator, multiply by $\frac{1 - z}{1 - z}$ .

$\frac{d z}{d u} = \frac{\frac{z}{1 - z} - z \cdot \frac{1 - z}{1 - z}}{u}$

Step 1.1.2.3.3

Simplify terms.

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Step 1.1.2.3.3.1

Combine $- z$ and $\frac{1 - z}{1 - z}$ .

$\frac{d z}{d u} = \frac{\frac{z}{1 - z} + \frac{- z (1 - z)}{1 - z}}{u}$

Step 1.1.2.3.3.2

Combine the numerators over the common denominator.

$\frac{d z}{d u} = \frac{\frac{z - z (1 - z)}{1 - z}}{u}$

Step 1.1.2.3.4

Simplify the numerator.

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Step 1.1.2.3.4.1

Factor $z$ out of $z - z (1 - z)$ .

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Step 1.1.2.3.4.1.1

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

$\frac{d z}{d u} = \frac{\frac{z^{1} - z (1 - z)}{1 - z}}{u}$

Step 1.1.2.3.4.1.2

Factor $z$ out of $z^{1}$ .

$\frac{d z}{d u} = \frac{\frac{z \cdot 1 - z (1 - z)}{1 - z}}{u}$

Step 1.1.2.3.4.1.3

Factor $z$ out of $- z (1 - z)$ .

$\frac{d z}{d u} = \frac{\frac{z \cdot 1 + z (- (1 - z))}{1 - z}}{u}$

Step 1.1.2.3.4.1.4

Factor $z$ out of $z \cdot 1 + z (- (1 - z))$ .

$\frac{d z}{d u} = \frac{\frac{z (1 - (1 - z))}{1 - z}}{u}$

Step 1.1.2.3.4.2

Apply the distributive property.

$\frac{d z}{d u} = \frac{\frac{z (1 - 1 \cdot 1 - - z)}{1 - z}}{u}$

Step 1.1.2.3.4.3

Multiply $- 1$ by $1$ .

$\frac{d z}{d u} = \frac{\frac{z (1 - 1 - - z)}{1 - z}}{u}$

Step 1.1.2.3.4.4

Multiply $- - z$ .

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Step 1.1.2.3.4.4.1

Multiply $- 1$ by $- 1$ .

$\frac{d z}{d u} = \frac{\frac{z (1 - 1 + 1 z)}{1 - z}}{u}$

Step 1.1.2.3.4.4.2

Multiply $z$ by $1$ .

$\frac{d z}{d u} = \frac{\frac{z (1 - 1 + z)}{1 - z}}{u}$

Step 1.1.2.3.4.5

Subtract $1$ from $1$ .

$\frac{d z}{d u} = \frac{\frac{z (0 + z)}{1 - z}}{u}$

Step 1.1.2.3.4.6

Add $0$ and $z$ .

$\frac{d z}{d u} = \frac{\frac{z \cdot z}{1 - z}}{u}$

Step 1.1.2.3.5

Simplify the numerator.

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Step 1.1.2.3.5.1

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

$\frac{d z}{d u} = \frac{\frac{z^{1} z}{1 - z}}{u}$

Step 1.1.2.3.5.2

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

$\frac{d z}{d u} = \frac{\frac{z^{1} z^{1}}{1 - z}}{u}$

Step 1.1.2.3.5.3

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

$\frac{d z}{d u} = \frac{\frac{z^{1 + 1}}{1 - z}}{u}$

Step 1.1.2.3.5.4

Add $1$ and $1$ .

$\frac{d z}{d u} = \frac{\frac{z^{2}}{1 - z}}{u}$

Step 1.1.2.3.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{d z}{d u} = \frac{z^{2}}{1 - z} \cdot \frac{1}{u}$

Step 1.1.2.3.7

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

$\frac{d z}{d u} = \frac{z^{2}}{(1 - z) u}$

Step 1.1.2.3.8

Reorder factors in $\frac{z^{2}}{(1 - z) u}$ .

$\frac{d z}{d u} = \frac{z^{2}}{u (1 - z)}$

Step 1.2

Regroup factors.

$\frac{d z}{d u} = \frac{z^{2}}{1 - z} \cdot \frac{1}{u}$

Step 1.3

Multiply both sides by $\frac{1 - z}{z^{2}}$ .

$\frac{1 - z}{z^{2}} \frac{d z}{d u} = \frac{1 - z}{z^{2}} (\frac{z^{2}}{1 - z} \cdot \frac{1}{u})$

Step 1.4

Simplify.

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Step 1.4.1

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

$\frac{1 - z}{z^{2}} \frac{d z}{d u} = \frac{1 - z}{z^{2}} \cdot \frac{z^{2}}{(1 - z) u}$

Step 1.4.2

Cancel the common factor of $1 - z$ .

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Step 1.4.2.1

Factor $1 - z$ out of $(1 - z) u$ .

$\frac{1 - z}{z^{2}} \frac{d z}{d u} = \frac{1 - z}{z^{2}} \cdot \frac{z^{2}}{(1 - z) (u)}$

Step 1.4.2.2

Cancel the common factor.

$\frac{1 - z}{z^{2}} \frac{d z}{d u} = \frac{1 - z}{z^{2}} \cdot \frac{z^{2}}{(1 - z) u}$

Step 1.4.2.3

Rewrite the expression.

$\frac{1 - z}{z^{2}} \frac{d z}{d u} = \frac{1}{z^{2}} \cdot \frac{z^{2}}{u}$

Step 1.4.3

Cancel the common factor of $z^{2}$ .

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Step 1.4.3.1

Cancel the common factor.

$\frac{1 - z}{z^{2}} \frac{d z}{d u} = \frac{1}{z^{2}} \cdot \frac{z^{2}}{u}$

Step 1.4.3.2

Rewrite the expression.

$\frac{1 - z}{z^{2}} \frac{d z}{d u} = \frac{1}{u}$

Step 1.5

Rewrite the equation.

$\frac{1 - z}{z^{2}} d z = \frac{1}{u} d u$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

$\int (1 - z) z^{2 \cdot - 1} d z = \int \frac{1}{u} d u$

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2

Multiply $(1 - z) z^{- 2}$ .

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

Multiply $z^{- 2}$ by $1$ .

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

Step 2.2.3.2

Multiply $z$ by $z^{- 2}$ by adding the exponents.

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Step 2.2.3.2.1

Move $z^{- 2}$ .

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

Step 2.2.3.2.2

Multiply $z^{- 2}$ by $z$ .

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Step 2.2.3.2.2.1

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

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

Step 2.2.3.2.2.2

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

$\int z^{- 2} - z^{- 2 + 1} d z = \int \frac{1}{u} d u$

Step 2.2.3.2.3

Add $- 2$ and $1$ .

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

Step 2.2.4

Split the single integral into multiple integrals.

$\int z^{- 2} d z + \int - z^{- 1} d z = \int \frac{1}{u} d u$

Step 2.2.5

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

$- z^{- 1} + C_{1} + \int - z^{- 1} d z = \int \frac{1}{u} d u$

Step 2.2.6

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

$- z^{- 1} + C_{1} - \int z^{- 1} d z = \int \frac{1}{u} d u$

Step 2.2.7

The integral of $z^{- 1}$ with respect to $z$ is $\ln (| z |)$ .

$- z^{- 1} + C_{1} - (\ln (| z |) + C_{2}) = \int \frac{1}{u} d u$

Step 2.2.8

Simplify.

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

Step 2.3

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

$- \frac{1}{z} - \ln (| z |) + C_{3} = \ln (| u |) + C_{4}$

Step 2.4

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

$- \frac{1}{z} - \ln (| z |) = \ln (| u |) + C$