Calculus Examples

$\int_{- 20}^{- 1} \frac{3}{e^{- z}} - \frac{1}{3 z} d z$

Step 1

Split the single integral into multiple integrals.

$\int_{- 20}^{- 1} \frac{3}{e^{- z}} d z + \int_{- 20}^{- 1} - \frac{1}{3 z} d z$

Step 2

Since $3$ is constant with respect to $z$ , move $3$ out of the integral.

$3 \int_{- 20}^{- 1} \frac{1}{e^{- z}} d z + \int_{- 20}^{- 1} - \frac{1}{3 z} d z$

Step 3

Simplify the expression.

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Step 3.1

Negate the exponent of $e^{- z}$ and move it out of the denominator.

$3 \int_{- 20}^{- 1} 1 {(e^{- z})}^{- 1} d z + \int_{- 20}^{- 1} - \frac{1}{3 z} d z$

Step 3.2

Simplify.

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Step 3.2.1

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

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Step 3.2.1.1

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

$3 \int_{- 20}^{- 1} 1 e^{- z \cdot - 1} d z + \int_{- 20}^{- 1} - \frac{1}{3 z} d z$

Step 3.2.1.2

Multiply $- z \cdot - 1$ .

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Step 3.2.1.2.1

Multiply $- 1$ by $- 1$ .

$3 \int_{- 20}^{- 1} 1 e^{1 z} d z + \int_{- 20}^{- 1} - \frac{1}{3 z} d z$

Step 3.2.1.2.2

Multiply $z$ by $1$ .

$3 \int_{- 20}^{- 1} 1 e^{z} d z + \int_{- 20}^{- 1} - \frac{1}{3 z} d z$

Step 3.2.2

Multiply $e^{z}$ by $1$ .

$3 \int_{- 20}^{- 1} e^{z} d z + \int_{- 20}^{- 1} - \frac{1}{3 z} d z$

Step 4

The integral of $e^{z}$ with respect to $z$ is $e^{z}$ .

$3 (e^{z}]_{- 20}^{- 1}) + \int_{- 20}^{- 1} - \frac{1}{3 z} d z$

Step 5

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

$3 (e^{z}]_{- 20}^{- 1}) - \int_{- 20}^{- 1} \frac{1}{3 z} d z$

Step 6

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

$3 (e^{z}]_{- 20}^{- 1}) - (\frac{1}{3} \int_{- 20}^{- 1} \frac{1}{z} d z)$

Step 7

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

$3 (e^{z}]_{- 20}^{- 1}) - \frac{1}{3} \ln (| z |)]_{- 20}^{- 1}$

Step 8

Simplify the answer.

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Step 8.1

Substitute and simplify.

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Step 8.1.1

Evaluate $e^{z}$ at $- 1$ and at $- 20$ .

$3 ((e^{- 1}) - e^{- 20}) - \frac{1}{3} \ln (| z |)]_{- 20}^{- 1}$

Step 8.1.2

Evaluate $\ln (| z |)$ at $- 1$ and at $- 20$ .

$3 ((e^{- 1}) - e^{- 20}) - \frac{1}{3} ((\ln (| - 1 |)) - \ln (| - 20 |))$

Step 8.1.3

Remove parentheses.

$3 (e^{- 1} - e^{- 20}) - \frac{1}{3} (\ln (| - 1 |) - \ln (| - 20 |))$

Step 8.2

Simplify.

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Step 8.2.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$3 (e^{- 1} - e^{- 20}) - \frac{1}{3} \ln (\frac{| - 1 |}{| - 20 |})$

Step 8.2.2

Combine $\ln (\frac{| - 1 |}{| - 20 |})$ and $\frac{1}{3}$ .

$3 (e^{- 1} - e^{- 20}) - \frac{\ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.2.3

To write $3 (e^{- 1} - e^{- 20})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 (e^{- 1} - e^{- 20}) \cdot \frac{3}{3} - \frac{\ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.2.4

Combine $3 (e^{- 1} - e^{- 20})$ and $\frac{3}{3}$ .

$\frac{3 (e^{- 1} - e^{- 20}) \cdot 3}{3} - \frac{\ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.2.5

Combine the numerators over the common denominator.

$\frac{3 (e^{- 1} - e^{- 20}) \cdot 3 - \ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.2.6

Multiply $3$ by $3$ .

$\frac{9 (e^{- 1} - e^{- 20}) - \ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.3

Simplify.

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Step 8.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{9 (e^{- 1} - \frac{1}{e^{20}}) - \ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{9 (\frac{1}{e} - \frac{1}{e^{20}}) - \ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.3.3

Apply the distributive property.

$\frac{9 \frac{1}{e} + 9 (- \frac{1}{e^{20}}) - \ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.3.4

Combine $9$ and $\frac{1}{e}$ .

$\frac{\frac{9}{e} + 9 (- \frac{1}{e^{20}}) - \ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.3.5

Multiply $9 (- \frac{1}{e^{20}})$ .

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Step 8.3.5.1

Multiply $- 1$ by $9$ .

$\frac{\frac{9}{e} - 9 \frac{1}{e^{20}} - \ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.3.5.2

Combine $- 9$ and $\frac{1}{e^{20}}$ .

$\frac{\frac{9}{e} + \frac{- 9}{e^{20}} - \ln (\frac{| - 1 |}{| - 20 |})}{3}$

Step 8.3.6

The absolute value is the distance between a number and zero. The distance between $- 1$ and $0$ is $1$ .

$\frac{\frac{9}{e} + \frac{- 9}{e^{20}} - \ln (\frac{1}{| - 20 |})}{3}$

Step 8.3.7

The absolute value is the distance between a number and zero. The distance between $- 20$ and $0$ is $20$ .

$\frac{\frac{9}{e} + \frac{- 9}{e^{20}} - \ln (\frac{1}{20})}{3}$

Step 8.3.8

Move the negative in front of the fraction.

$\frac{\frac{9}{e} - \frac{9}{e^{20}} - \ln (\frac{1}{20})}{3}$

Step 8.3.9

Simplify the numerator.

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Step 8.3.9.1

To write $\frac{9}{e}$ as a fraction with a common denominator, multiply by $\frac{e^{19}}{e^{19}}$ .

$\frac{\frac{9}{e} \cdot \frac{e^{19}}{e^{19}} - \frac{9}{e^{20}} - \ln (\frac{1}{20})}{3}$

Step 8.3.9.2

Write each expression with a common denominator of $e^{20}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.9.2.1

Multiply $\frac{9}{e}$ by $\frac{e^{19}}{e^{19}}$ .

$\frac{\frac{9 e^{19}}{e e^{19}} - \frac{9}{e^{20}} - \ln (\frac{1}{20})}{3}$

Step 8.3.9.2.2

Multiply $e$ by $e^{19}$ by adding the exponents.

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Step 8.3.9.2.2.1

Multiply $e$ by $e^{19}$ .

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Step 8.3.9.2.2.1.1

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

$\frac{\frac{9 e^{19}}{e^{1} e^{19}} - \frac{9}{e^{20}} - \ln (\frac{1}{20})}{3}$

Step 8.3.9.2.2.1.2

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

$\frac{\frac{9 e^{19}}{e^{1 + 19}} - \frac{9}{e^{20}} - \ln (\frac{1}{20})}{3}$

Step 8.3.9.2.2.2

Add $1$ and $19$ .

$\frac{\frac{9 e^{19}}{e^{20}} - \frac{9}{e^{20}} - \ln (\frac{1}{20})}{3}$

Step 8.3.9.3

Combine the numerators over the common denominator.

$\frac{\frac{9 e^{19} - 9}{e^{20}} - \ln (\frac{1}{20})}{3}$

Step 8.3.9.4

To write $- \ln (\frac{1}{20})$ as a fraction with a common denominator, multiply by $\frac{e^{20}}{e^{20}}$ .

$\frac{\frac{9 e^{19} - 9}{e^{20}} - \ln (\frac{1}{20}) \cdot \frac{e^{20}}{e^{20}}}{3}$

Step 8.3.9.5

Combine $- \ln (\frac{1}{20})$ and $\frac{e^{20}}{e^{20}}$ .

$\frac{\frac{9 e^{19} - 9}{e^{20}} + \frac{- \ln (\frac{1}{20}) e^{20}}{e^{20}}}{3}$

Step 8.3.9.6

Combine the numerators over the common denominator.

$\frac{\frac{9 e^{19} - 9 - \ln (\frac{1}{20}) e^{20}}{e^{20}}}{3}$

Step 8.3.10

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 e^{19} - 9 - \ln (\frac{1}{20}) e^{20}}{e^{20}} \cdot \frac{1}{3}$

Step 8.3.11

Multiply $\frac{9 e^{19} - 9 - \ln (\frac{1}{20}) e^{20}}{e^{20}}$ by $\frac{1}{3}$ .

$\frac{9 e^{19} - 9 - \ln (\frac{1}{20}) e^{20}}{e^{20} \cdot 3}$

Step 8.3.12

Move $3$ to the left of $e^{20}$ .

$\frac{9 e^{19} - 9 - \ln (\frac{1}{20}) e^{20}}{3 \cdot e^{20}}$

Step 8.3.13

Reorder factors in $\frac{9 e^{19} - 9 - \ln (\frac{1}{20}) e^{20}}{3 e^{20}}$ .

$\frac{9 e^{19} - 9 - e^{20} \ln (\frac{1}{20})}{3 e^{20}}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{9 e^{19} - 9 - e^{20} \ln (\frac{1}{20})}{3 e^{20}}$

Decimal Form:

$2.10221574 \dots$

Step 10