Calculus Examples

Evaluate the Integral integral of xcos(2x)^2 with respect to x
Step 1
Integrate by parts using the formula , where and .
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
Since is constant with respect to , move out of the integral.
Step 6
Let . Then , so . Rewrite using and .
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Step 6.1
Let . Find .
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Step 6.1.1
Differentiate .
Step 6.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.1.3
Differentiate using the Power Rule which states that is where .
Step 6.1.4
Multiply by .
Step 6.2
Rewrite the problem using and .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Simplify.
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Step 8.1
Combine and .
Step 8.2
Combine and .
Step 9
The integral of with respect to is .
Step 10
Simplify.
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Step 10.1
Simplify.
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Step 10.1.1
Multiply by .
Step 10.1.2
Multiply by .
Step 10.1.3
Combine and .
Step 10.2
Simplify.
Step 10.3
Simplify.
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Step 10.3.1
Multiply by .
Step 10.3.2
Multiply by .
Step 10.3.3
Combine and .
Step 10.3.4
To write as a fraction with a common denominator, multiply by .
Step 10.3.5
Combine and .
Step 10.3.6
Combine the numerators over the common denominator.
Step 10.3.7
Multiply by .
Step 11
Replace all occurrences of with .
Step 12
Simplify.
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Step 12.1
Apply the distributive property.
Step 12.2
Cancel the common factor of .
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Step 12.2.1
Factor out of .
Step 12.2.2
Cancel the common factor.
Step 12.2.3
Rewrite the expression.
Step 12.3
Cancel the common factor of .
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Step 12.3.1
Move the leading negative in into the numerator.
Step 12.3.2
Factor out of .
Step 12.3.3
Factor out of .
Step 12.3.4
Cancel the common factor.
Step 12.3.5
Rewrite the expression.
Step 12.4
Simplify each term.
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Step 12.4.1
Move the negative in front of the fraction.
Step 12.4.2
Multiply .
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Step 12.4.2.1
Multiply by .
Step 12.4.2.2
Multiply by .
Step 13
Reorder terms.