Calculus Examples

Evaluate the Integral integral of xe^(-2x) 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
Rewrite the problem using and .
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
Simplify.
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Step 11.1
Rewrite as .
Step 11.2
Simplify.
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Step 11.2.1
Combine and .
Step 11.2.2
Combine and .
Step 12
Replace all occurrences of with .
Step 13
Combine and .
Step 14
Reorder terms.