Calculus Examples

Evaluate the Integral integral of x^2cos(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
Combine and .
Step 4.2
Cancel the common factor of and .
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Step 4.2.1
Factor out of .
Step 4.2.2
Cancel the common factors.
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Step 4.2.2.1
Factor out of .
Step 4.2.2.2
Cancel the common factor.
Step 4.2.2.3
Rewrite the expression.
Step 5
Integrate by parts using the formula , where and .
Step 6
Simplify.
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Step 6.1
Combine and .
Step 6.2
Combine and .
Step 6.3
Combine and .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Simplify.
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Step 8.1
Multiply by .
Step 8.2
Multiply by .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
Let . Then , so . Rewrite using and .
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Step 10.1
Let . Find .
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Step 10.1.1
Differentiate .
Step 10.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 10.1.3
Differentiate using the Power Rule which states that is where .
Step 10.1.4
Multiply by .
Step 10.2
Rewrite the problem using and .
Step 11
Combine and .
Step 12
Since is constant with respect to , move out of the integral.
Step 13
Simplify.
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Step 13.1
Multiply by .
Step 13.2
Multiply by .
Step 14
The integral of with respect to is .
Step 15
Simplify.
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Step 15.1
Rewrite as .
Step 15.2
Simplify.
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Step 15.2.1
Combine and .
Step 15.2.2
Combine and .
Step 15.2.3
Combine and .
Step 15.2.4
Combine and .
Step 15.2.5
Combine and .
Step 15.2.6
To write as a fraction with a common denominator, multiply by .
Step 15.2.7
Combine and .
Step 15.2.8
Combine the numerators over the common denominator.
Step 15.2.9
Multiply by .
Step 15.2.10
Combine and .
Step 15.2.11
Cancel the common factor of and .
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Step 15.2.11.1
Factor out of .
Step 15.2.11.2
Cancel the common factors.
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Step 15.2.11.2.1
Factor out of .
Step 15.2.11.2.2
Cancel the common factor.
Step 15.2.11.2.3
Rewrite the expression.
Step 15.2.11.2.4
Divide by .
Step 16
Replace all occurrences of with .
Step 17
Simplify.
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Step 17.1
Apply the distributive property.
Step 17.2
Cancel the common factor of .
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Step 17.2.1
Move the leading negative in into the numerator.
Step 17.2.2
Factor out of .
Step 17.2.3
Factor out of .
Step 17.2.4
Cancel the common factor.
Step 17.2.5
Rewrite the expression.
Step 17.3
Cancel the common factor of .
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Step 17.3.1
Factor out of .
Step 17.3.2
Factor out of .
Step 17.3.3
Cancel the common factor.
Step 17.3.4
Rewrite the expression.
Step 17.4
Simplify each term.
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Step 17.4.1
Move the negative in front of the fraction.
Step 17.4.2
Multiply .
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Step 17.4.2.1
Multiply by .
Step 17.4.2.2
Multiply by .
Step 17.5
Reorder factors in .
Step 18
Simplify.
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Step 18.1
Simplify the numerator.
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Step 18.1.1
To write as a fraction with a common denominator, multiply by .
Step 18.1.2
Combine and .
Step 18.1.3
Combine the numerators over the common denominator.
Step 18.1.4
To write as a fraction with a common denominator, multiply by .
Step 18.1.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 18.1.5.1
Multiply by .
Step 18.1.5.2
Multiply by .
Step 18.1.6
Combine the numerators over the common denominator.
Step 18.1.7
Rewrite in a factored form.
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Step 18.1.7.1
Move to the left of .
Step 18.1.7.2
Move to the left of .
Step 18.2
Multiply the numerator by the reciprocal of the denominator.
Step 18.3
Multiply .
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Step 18.3.1
Multiply by .
Step 18.3.2
Multiply by .
Step 18.4
Reorder terms.