Calculus Examples

Evaluate the Integral integral of (x^2+5x+6)cos(2x) with respect to x
Step 1
Apply the distributive property.
Step 2
Split the single integral into multiple integrals.
Step 3
Integrate by parts using the formula , where and .
Step 4
Simplify.
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Step 4.1
Combine and .
Step 4.2
Combine and .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Simplify.
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Step 6.1
Combine and .
Step 6.2
Cancel the common factor of .
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Step 6.2.1
Cancel the common factor.
Step 6.2.2
Rewrite the expression.
Step 6.3
Multiply by .
Step 7
Integrate by parts using the formula , where and .
Step 8
Simplify.
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Step 8.1
Combine and .
Step 8.2
Combine and .
Step 8.3
Combine and .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
Simplify.
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Step 10.1
Multiply by .
Step 10.2
Multiply by .
Step 11
Since is constant with respect to , move out of the integral.
Step 12
Let . Then , so . Rewrite using and .
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Step 12.1
Let . Find .
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Step 12.1.1
Differentiate .
Step 12.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 12.1.3
Differentiate using the Power Rule which states that is where .
Step 12.1.4
Multiply by .
Step 12.2
Rewrite the problem using and .
Step 13
Combine and .
Step 14
Since is constant with respect to , move out of the integral.
Step 15
Simplify.
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Step 15.1
Multiply by .
Step 15.2
Multiply by .
Step 16
The integral of with respect to is .
Step 17
Since is constant with respect to , move out of the integral.
Step 18
Integrate by parts using the formula , where and .
Step 19
Simplify.
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Step 19.1
Combine and .
Step 19.2
Combine and .
Step 19.3
Combine and .
Step 20
Since is constant with respect to , move out of the integral.
Step 21
Let . Then , so . Rewrite using and .
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Step 21.1
Let . Find .
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Step 21.1.1
Differentiate .
Step 21.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 21.1.3
Differentiate using the Power Rule which states that is where .
Step 21.1.4
Multiply by .
Step 21.2
Rewrite the problem using and .
Step 22
Combine and .
Step 23
Since is constant with respect to , move out of the integral.
Step 24
Simplify.
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Step 24.1
Multiply by .
Step 24.2
Multiply by .
Step 25
The integral of with respect to is .
Step 26
Since is constant with respect to , move out of the integral.
Step 27
Let . Then , so . Rewrite using and .
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Step 27.1
Let . Find .
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Step 27.1.1
Differentiate .
Step 27.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 27.1.3
Differentiate using the Power Rule which states that is where .
Step 27.1.4
Multiply by .
Step 27.2
Rewrite the problem using and .
Step 28
Combine and .
Step 29
Since is constant with respect to , move out of the integral.
Step 30
Simplify.
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Step 30.1
Combine and .
Step 30.2
Cancel the common factor of and .
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Step 30.2.1
Factor out of .
Step 30.2.2
Cancel the common factors.
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Step 30.2.2.1
Factor out of .
Step 30.2.2.2
Cancel the common factor.
Step 30.2.2.3
Rewrite the expression.
Step 30.2.2.4
Divide by .
Step 31
The integral of with respect to is .
Step 32
Simplify.
Step 33
Substitute back in for each integration substitution variable.
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Step 33.1
Replace all occurrences of with .
Step 33.2
Replace all occurrences of with .
Step 33.3
Replace all occurrences of with .
Step 34
Reorder terms.