Calculus Examples

Evaluate the Integral integral from 0 to 2 of |z-z^2| with respect to z
Step 1
Split up the integral depending on where is positive and negative.
Step 2
Split the single integral into multiple integrals.
Step 3
By the Power Rule, the integral of with respect to is .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Combine and .
Step 7
Split the single integral into multiple integrals.
Step 8
Since is constant with respect to , move out of the integral.
Step 9
By the Power Rule, the integral of with respect to is .
Step 10
Combine and .
Step 11
By the Power Rule, the integral of with respect to is .
Step 12
Substitute and simplify.
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Step 12.1
Evaluate at and at .
Step 12.2
Evaluate at and at .
Step 12.3
Evaluate at and at .
Step 12.4
Evaluate at and at .
Step 12.5
Simplify.
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Step 12.5.1
One to any power is one.
Step 12.5.2
Multiply by .
Step 12.5.3
Raising to any positive power yields .
Step 12.5.4
Multiply by .
Step 12.5.5
Multiply by .
Step 12.5.6
Add and .
Step 12.5.7
One to any power is one.
Step 12.5.8
Raising to any positive power yields .
Step 12.5.9
Cancel the common factor of and .
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Step 12.5.9.1
Factor out of .
Step 12.5.9.2
Cancel the common factors.
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Step 12.5.9.2.1
Factor out of .
Step 12.5.9.2.2
Cancel the common factor.
Step 12.5.9.2.3
Rewrite the expression.
Step 12.5.9.2.4
Divide by .
Step 12.5.10
Multiply by .
Step 12.5.11
Add and .
Step 12.5.12
To write as a fraction with a common denominator, multiply by .
Step 12.5.13
To write as a fraction with a common denominator, multiply by .
Step 12.5.14
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 12.5.14.1
Multiply by .
Step 12.5.14.2
Multiply by .
Step 12.5.14.3
Multiply by .
Step 12.5.14.4
Multiply by .
Step 12.5.15
Combine the numerators over the common denominator.
Step 12.5.16
Subtract from .
Step 12.5.17
Raise to the power of .
Step 12.5.18
Cancel the common factor of and .
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Step 12.5.18.1
Factor out of .
Step 12.5.18.2
Cancel the common factors.
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Step 12.5.18.2.1
Factor out of .
Step 12.5.18.2.2
Cancel the common factor.
Step 12.5.18.2.3
Rewrite the expression.
Step 12.5.18.2.4
Divide by .
Step 12.5.19
One to any power is one.
Step 12.5.20
To write as a fraction with a common denominator, multiply by .
Step 12.5.21
Combine and .
Step 12.5.22
Combine the numerators over the common denominator.
Step 12.5.23
Simplify the numerator.
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Step 12.5.23.1
Multiply by .
Step 12.5.23.2
Subtract from .
Step 12.5.24
To write as a fraction with a common denominator, multiply by .
Step 12.5.25
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 12.5.25.1
Multiply by .
Step 12.5.25.2
Multiply by .
Step 12.5.26
Combine the numerators over the common denominator.
Step 12.5.27
Simplify the numerator.
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Step 12.5.27.1
Multiply by .
Step 12.5.27.2
Subtract from .
Step 12.5.28
Cancel the common factor of and .
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Step 12.5.28.1
Factor out of .
Step 12.5.28.2
Cancel the common factors.
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Step 12.5.28.2.1
Factor out of .
Step 12.5.28.2.2
Cancel the common factor.
Step 12.5.28.2.3
Rewrite the expression.
Step 12.5.29
Move the negative in front of the fraction.
Step 12.5.30
Raise to the power of .
Step 12.5.31
Combine and .
Step 12.5.32
One to any power is one.
Step 12.5.33
Multiply by .
Step 12.5.34
Combine the numerators over the common denominator.
Step 12.5.35
Subtract from .
Step 12.5.36
Combine the numerators over the common denominator.
Step 12.5.37
Add and .
Step 12.5.38
Cancel the common factor of .
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Step 12.5.38.1
Cancel the common factor.
Step 12.5.38.2
Rewrite the expression.
Step 13