Calculus Examples

Evaluate the Integral integral from -8 to 0 of xe^(-4x) with respect to x
Step 1
Integrate by parts using the formula , where and .
Step 2
Simplify.
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Step 2.1
Combine and .
Step 2.2
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Simplify.
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Step 4.1
Multiply by .
Step 4.2
Multiply by .
Step 5
Let . Then , so . Rewrite using and .
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Step 5.1
Let . Find .
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Step 5.1.1
Differentiate .
Step 5.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3
Differentiate using the Power Rule which states that is where .
Step 5.1.4
Multiply by .
Step 5.2
Substitute the lower limit in for in .
Step 5.3
Multiply by .
Step 5.4
Substitute the upper limit in for in .
Step 5.5
Multiply by .
Step 5.6
The values found for and will be used to evaluate the definite integral.
Step 5.7
Rewrite the problem using , , and the new limits of integration.
Step 6
Simplify.
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Step 6.1
Move the negative in front of the fraction.
Step 6.2
Combine and .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Since is constant with respect to , move out of the integral.
Step 9
Simplify.
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Step 9.1
Multiply by .
Step 9.2
Multiply by .
Step 10
The integral of with respect to is .
Step 11
Substitute and simplify.
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Step 11.1
Evaluate at and at .
Step 11.2
Evaluate at and at .
Step 11.3
Simplify.
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Step 11.3.1
Multiply by .
Step 11.3.2
Anything raised to is .
Step 11.3.3
Multiply by .
Step 11.3.4
Cancel the common factor of and .
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Step 11.3.4.1
Factor out of .
Step 11.3.4.2
Cancel the common factors.
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Step 11.3.4.2.1
Factor out of .
Step 11.3.4.2.2
Cancel the common factor.
Step 11.3.4.2.3
Rewrite the expression.
Step 11.3.4.2.4
Divide by .
Step 11.3.5
Multiply by .
Step 11.3.6
Multiply by .
Step 11.3.7
Cancel the common factor of and .
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Step 11.3.7.1
Factor out of .
Step 11.3.7.2
Cancel the common factors.
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Step 11.3.7.2.1
Factor out of .
Step 11.3.7.2.2
Cancel the common factor.
Step 11.3.7.2.3
Rewrite the expression.
Step 11.3.7.2.4
Divide by .
Step 11.3.8
Subtract from .
Step 11.3.9
Anything raised to is .
Step 12
Simplify.
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Step 12.1
Simplify each term.
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Step 12.1.1
Apply the distributive property.
Step 12.1.2
Multiply by .
Step 12.1.3
Multiply .
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Step 12.1.3.1
Multiply by .
Step 12.1.3.2
Multiply by .
Step 12.1.3.3
Combine and .
Step 12.2
To write as a fraction with a common denominator, multiply by .
Step 12.3
Combine and .
Step 12.4
Combine the numerators over the common denominator.
Step 12.5
Combine the numerators over the common denominator.
Step 12.6
Multiply by .
Step 12.7
Add and .
Step 13
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 14