Calculus Examples

Evaluate the Integral integral from 0 to 3 of |2x-3| with respect to x
Step 1
Split up the integral depending on where is positive and negative.
Step 2
Split the single integral into multiple integrals.
Step 3
Since is constant with respect to , move out of the integral.
Step 4
By the Power Rule, the integral of with respect to is .
Step 5
Combine and .
Step 6
Apply the constant rule.
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
Apply the constant rule.
Step 12
Simplify the answer.
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Step 12.1
Substitute and simplify.
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Step 12.1.1
Evaluate at and at .
Step 12.1.2
Evaluate at and at .
Step 12.1.3
Evaluate at and at .
Step 12.1.4
Evaluate at and at .
Step 12.1.5
Simplify.
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Step 12.1.5.1
Raising to any positive power yields .
Step 12.1.5.2
Cancel the common factor of and .
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Step 12.1.5.2.1
Factor out of .
Step 12.1.5.2.2
Cancel the common factors.
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Step 12.1.5.2.2.1
Factor out of .
Step 12.1.5.2.2.2
Cancel the common factor.
Step 12.1.5.2.2.3
Rewrite the expression.
Step 12.1.5.2.2.4
Divide by .
Step 12.1.5.3
Multiply by .
Step 12.1.5.4
Add and .
Step 12.1.5.5
Combine and .
Step 12.1.5.6
Multiply by .
Step 12.1.5.7
Multiply by .
Step 12.1.5.8
Add and .
Step 12.1.5.9
Raise to the power of .
Step 12.1.5.10
To write as a fraction with a common denominator, multiply by .
Step 12.1.5.11
Combine and .
Step 12.1.5.12
Combine the numerators over the common denominator.
Step 12.1.5.13
Multiply by .
Step 12.1.5.14
Multiply by .
Step 12.1.5.15
Combine and .
Step 12.1.5.16
Multiply by .
Step 12.1.5.17
To write as a fraction with a common denominator, multiply by .
Step 12.1.5.18
Combine and .
Step 12.1.5.19
Combine the numerators over the common denominator.
Step 12.1.5.20
Simplify the numerator.
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Step 12.1.5.20.1
Multiply by .
Step 12.1.5.20.2
Add and .
Step 12.1.5.21
Move the negative in front of the fraction.
Step 12.2
Simplify.
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Step 12.2.1
Combine the numerators over the common denominator.
Step 12.2.2
Simplify each term.
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Step 12.2.2.1
Combine the numerators over the common denominator.
Step 12.2.2.2
Simplify each term.
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Step 12.2.2.2.1
Apply the product rule to .
Step 12.2.2.2.2
Raise to the power of .
Step 12.2.2.2.3
Raise to the power of .
Step 12.2.2.3
To write as a fraction with a common denominator, multiply by .
Step 12.2.2.4
Combine and .
Step 12.2.2.5
Combine the numerators over the common denominator.
Step 12.2.2.6
Simplify the numerator.
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Step 12.2.2.6.1
Multiply by .
Step 12.2.2.6.2
Subtract from .
Step 12.2.2.7
Cancel the common factor of .
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Step 12.2.2.7.1
Factor out of .
Step 12.2.2.7.2
Cancel the common factor.
Step 12.2.2.7.3
Rewrite the expression.
Step 12.2.2.8
Combine and .
Step 12.2.2.9
Multiply by .
Step 12.2.2.10
Cancel the common factor of and .
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Step 12.2.2.10.1
Factor out of .
Step 12.2.2.10.2
Cancel the common factors.
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Step 12.2.2.10.2.1
Factor out of .
Step 12.2.2.10.2.2
Cancel the common factor.
Step 12.2.2.10.2.3
Rewrite the expression.
Step 12.2.3
Find the common denominator.
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Step 12.2.3.1
Write as a fraction with denominator .
Step 12.2.3.2
Multiply by .
Step 12.2.3.3
Multiply by .
Step 12.2.3.4
Write as a fraction with denominator .
Step 12.2.3.5
Multiply by .
Step 12.2.3.6
Multiply by .
Step 12.2.4
Combine the numerators over the common denominator.
Step 12.2.5
Simplify each term.
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Step 12.2.5.1
Multiply by .
Step 12.2.5.2
Multiply by .
Step 12.2.6
Add and .
Step 12.2.7
Subtract from .
Step 12.2.8
Simplify each term.
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Step 12.2.8.1
Cancel the common factor of .
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Step 12.2.8.1.1
Factor out of .
Step 12.2.8.1.2
Cancel the common factor.
Step 12.2.8.1.3
Rewrite the expression.
Step 12.2.8.2
Apply the product rule to .
Step 12.2.8.3
Raise to the power of .
Step 12.2.8.4
Raise to the power of .
Step 12.2.8.5
Multiply the numerator by the reciprocal of the denominator.
Step 12.2.8.6
Multiply .
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Step 12.2.8.6.1
Multiply by .
Step 12.2.8.6.2
Multiply by .
Step 12.2.9
Combine the numerators over the common denominator.
Step 12.2.10
Add and .
Step 12.2.11
Cancel the common factor of and .
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Step 12.2.11.1
Factor out of .
Step 12.2.11.2
Cancel the common factors.
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Step 12.2.11.2.1
Factor out of .
Step 12.2.11.2.2
Cancel the common factor.
Step 12.2.11.2.3
Rewrite the expression.
Step 13
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Mixed Number Form:
Step 14