Calculus Examples

Evaluate the Integral integral from -20 to -1 of 3/(e^(-z))-1/(3z) with respect to z
-1-203e-z-13zdz1203ez13zdz
Step 1
Split the single integral into multiple integrals.
-1-203e-zdz+-1-20-13zdz
Step 2
Since 3 is constant with respect to z, move 3 out of the integral.
3-1-201e-zdz+-1-20-13zdz
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.
3-1-201(e-z)-1dz+-1-20-13zdz
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, (am)n=amn.
3-1-201e-z-1dz+-1-20-13zdz
Step 3.2.1.2
Multiply -z-1.
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Step 3.2.1.2.1
Multiply -1 by -1.
3-1-201e1zdz+-1-20-13zdz
Step 3.2.1.2.2
Multiply z by 1.
3-1-201ezdz+-1-20-13zdz
3-1-201ezdz+-1-20-13zdz
3-1-201ezdz+-1-20-13zdz
Step 3.2.2
Multiply ez by 1.
3-1-20ezdz+-1-20-13zdz
3-1-20ezdz+-1-20-13zdz
3-1-20ezdz+-1-20-13zdz
Step 4
The integral of ez with respect to z is ez.
3(ez]-1-20)+-1-20-13zdz
Step 5
Since -1 is constant with respect to z, move -1 out of the integral.
3(ez]-1-20)--1-2013zdz
Step 6
Since 13 is constant with respect to z, move 13 out of the integral.
3(ez]-1-20)-(13-1-201zdz)
Step 7
The integral of 1z with respect to z is ln(|z|).
3(ez]-1-20)-13ln(|z|)]-1-20
Step 8
Simplify the answer.
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Step 8.1
Substitute and simplify.
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Step 8.1.1
Evaluate ez at -1 and at -20.
3((e-1)-e-20)-13ln(|z|)]-1-20
Step 8.1.2
Evaluate ln(|z|) at -1 and at -20.
3((e-1)-e-20)-13((ln(|-1|))-ln(|-20|))
Step 8.1.3
Remove parentheses.
3(e-1-e-20)-13(ln(|-1|)-ln(|-20|))
3(e-1-e-20)-13(ln(|-1|)-ln(|-20|))
Step 8.2
Simplify.
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Step 8.2.1
Use the quotient property of logarithms, logb(x)-logb(y)=logb(xy).
3(e-1-e-20)-13ln(|-1||-20|)
Step 8.2.2
Combine ln(|-1||-20|) and 13.
3(e-1-e-20)-ln(|-1||-20|)3
Step 8.2.3
To write 3(e-1-e-20) as a fraction with a common denominator, multiply by 33.
3(e-1-e-20)33-ln(|-1||-20|)3
Step 8.2.4
Combine 3(e-1-e-20) and 33.
3(e-1-e-20)33-ln(|-1||-20|)3
Step 8.2.5
Combine the numerators over the common denominator.
3(e-1-e-20)3-ln(|-1||-20|)3
Step 8.2.6
Multiply 3 by 3.
9(e-1-e-20)-ln(|-1||-20|)3
9(e-1-e-20)-ln(|-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=1bn.
9(e-1-1e20)-ln(|-1||-20|)3
Step 8.3.2
Rewrite the expression using the negative exponent rule b-n=1bn.
9(1e-1e20)-ln(|-1||-20|)3
Step 8.3.3
Apply the distributive property.
91e+9(-1e20)-ln(|-1||-20|)3
Step 8.3.4
Combine 9 and 1e.
9e+9(-1e20)-ln(|-1||-20|)3
Step 8.3.5
Multiply 9(-1e20).
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Step 8.3.5.1
Multiply -1 by 9.
9e-91e20-ln(|-1||-20|)3
Step 8.3.5.2
Combine -9 and 1e20.
9e+-9e20-ln(|-1||-20|)3
9e+-9e20-ln(|-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.
9e+-9e20-ln(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.
9e+-9e20-ln(120)3
Step 8.3.8
Move the negative in front of the fraction.
9e-9e20-ln(120)3
Step 8.3.9
Simplify the numerator.
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Step 8.3.9.1
To write 9e as a fraction with a common denominator, multiply by e19e19.
9ee19e19-9e20-ln(120)3
Step 8.3.9.2
Write each expression with a common denominator of e20, by multiplying each by an appropriate factor of 1.
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Step 8.3.9.2.1
Multiply 9e by e19e19.
9e19ee19-9e20-ln(120)3
Step 8.3.9.2.2
Multiply e by e19 by adding the exponents.
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Step 8.3.9.2.2.1
Multiply e by e19.
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Step 8.3.9.2.2.1.1
Raise e to the power of 1.
9e19e1e19-9e20-ln(120)3
Step 8.3.9.2.2.1.2
Use the power rule aman=am+n to combine exponents.
9e19e1+19-9e20-ln(120)3
9e19e1+19-9e20-ln(120)3
Step 8.3.9.2.2.2
Add 1 and 19.
9e19e20-9e20-ln(120)3
9e19e20-9e20-ln(120)3
9e19e20-9e20-ln(120)3
Step 8.3.9.3
Combine the numerators over the common denominator.
9e19-9e20-ln(120)3
Step 8.3.9.4
To write -ln(120) as a fraction with a common denominator, multiply by e20e20.
9e19-9e20-ln(120)e20e203
Step 8.3.9.5
Combine -ln(120) and e20e20.
9e19-9e20+-ln(120)e20e203
Step 8.3.9.6
Combine the numerators over the common denominator.
9e19-9-ln(120)e20e203
9e19-9-ln(120)e20e203
Step 8.3.10
Multiply the numerator by the reciprocal of the denominator.
9e19-9-ln(120)e20e2013
Step 8.3.11
Multiply 9e19-9-ln(120)e20e20 by 13.
9e19-9-ln(120)e20e203
Step 8.3.12
Move 3 to the left of e20.
9e19-9-ln(120)e203e20
Step 8.3.13
Reorder factors in 9e19-9-ln(120)e203e20.
9e19-9-e20ln(120)3e20
9e19-9-e20ln(120)3e20
9e19-9-e20ln(120)3e20
Step 9
The result can be shown in multiple forms.
Exact Form:
9e19-9-e20ln(120)3e20
Decimal Form:
2.10221574
Step 10
 [x2  12  π  xdx ]