Calculus Examples

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

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

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

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

Step 2

Since

3

$3$ is constant with respect to

z

$z$ , move

3

$3$ out of the integral.

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

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

Step 3

Simplify the expression.

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Step 3.1

Negate the exponent of

e^{- z}

$e^{- z}$ 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

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

Step 3.2

Simplify.

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Step 3.2.1

Multiply the exponents in

{(e^{- z})}^{- 1}

${(e^{- z})}^{- 1}$ .

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Step 3.2.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 3.2.1.2

Multiply

- z \cdot - 1

$- z \cdot - 1$ .

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Step 3.2.1.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 3.2.1.2.2

Multiply

z

$z$ by

1

$1$ .

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

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

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

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

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

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

Step 3.2.2

Multiply

e^{z}

$e^{z}$ by

1

$1$ .

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

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

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

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

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

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

Step 4

The integral of

e^{z}

$e^{z}$ with respect to

z

$z$ is

e^{z}

$e^{z}$ .

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

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

Step 5

Since

- 1

$- 1$ is constant with respect to

z

$z$ , move

- 1

$- 1$ out of the integral.

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

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

Step 6

Since

\frac{1}{3}

$\frac{1}{3}$ is constant with respect to

z

$z$ , move

\frac{1}{3}

$\frac{1}{3}$ out of the integral.

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

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

Step 7

The integral of

\frac{1}{z}

$\frac{1}{z}$ with respect to

z

$z$ is

ln (| z |)

$\ln (| z |)$ .

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

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

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}

$e^{z}$ at

- 1

$- 1$ and at

- 20

$- 20$ .

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

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

Step 8.1.2

Evaluate

ln (| z |)

$\ln (| z |)$ at

- 1

$- 1$ and at

- 20

$- 20$ .

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

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

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

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

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})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

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

Step 8.2.2

Combine

ln (\frac{| - 1 |}{| - 20 |})

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

\frac{1}{3}

$\frac{1}{3}$ .

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

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

Step 8.2.3

To write

3 (e^{- 1} - e^{- 20})

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

\frac{3}{3}

$\frac{3}{3}$ .

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

$3 (e^{- 1} - e^{- 20})$ and

\frac{3}{3}

$\frac{3}{3}$ .

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