Calculus Examples

Evaluate the Integral integral from 0 to 2pi of t^2sin(2t) with respect to t
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
Combine and .
Step 4.3
Cancel the common factor of and .
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Step 4.3.1
Factor out of .
Step 4.3.2
Cancel the common factors.
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Step 4.3.2.1
Factor out of .
Step 4.3.2.2
Cancel the common factor.
Step 4.3.2.3
Rewrite the expression.
Step 4.3.2.4
Divide by .
Step 4.4
Multiply by .
Step 4.5
Multiply by .
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 7
Since is constant with respect to , move out of the integral.
Step 8
Let . Then , so . Rewrite using and .
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Step 8.1
Let . Find .
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Step 8.1.1
Differentiate .
Step 8.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 8.1.3
Differentiate using the Power Rule which states that is where .
Step 8.1.4
Multiply by .
Step 8.2
Substitute the lower limit in for in .
Step 8.3
Multiply by .
Step 8.4
Substitute the upper limit in for in .
Step 8.5
Multiply by .
Step 8.6
The values found for and will be used to evaluate the definite integral.
Step 8.7
Rewrite the problem using , , and the new limits of integration.
Step 9
Combine and .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Simplify.
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Step 11.1
Multiply by .
Step 11.2
Multiply by .
Step 12
The integral of with respect to is .
Step 13
Combine and .
Step 14
Substitute and simplify.
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Step 14.1
Evaluate at and at .
Step 14.2
Evaluate at and at .
Step 14.3
Simplify.
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Step 14.3.1
Factor out of .
Step 14.3.2
Apply the product rule to .
Step 14.3.3
Raise to the power of .
Step 14.3.4
Multiply by .
Step 14.3.5
Cancel the common factor of and .
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Step 14.3.5.1
Factor out of .
Step 14.3.5.2
Cancel the common factors.
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Step 14.3.5.2.1
Factor out of .
Step 14.3.5.2.2
Cancel the common factor.
Step 14.3.5.2.3
Rewrite the expression.
Step 14.3.5.2.4
Divide by .
Step 14.3.6
Multiply by .
Step 14.3.7
Multiply by .
Step 14.3.8
Cancel the common factor of .
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Step 14.3.8.1
Cancel the common factor.
Step 14.3.8.2
Divide by .
Step 14.3.9
Raising to any positive power yields .
Step 14.3.10
Multiply by .
Step 14.3.11
Multiply by .
Step 14.3.12
Cancel the common factor of and .
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Step 14.3.12.1
Factor out of .
Step 14.3.12.2
Cancel the common factors.
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Step 14.3.12.2.1
Factor out of .
Step 14.3.12.2.2
Cancel the common factor.
Step 14.3.12.2.3
Rewrite the expression.
Step 14.3.12.2.4
Divide by .
Step 14.3.13
Multiply by .
Step 14.3.14
Multiply by .
Step 14.3.15
Multiply by .
Step 14.3.16
Cancel the common factor of and .
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Step 14.3.16.1
Factor out of .
Step 14.3.16.2
Cancel the common factors.
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Step 14.3.16.2.1
Factor out of .
Step 14.3.16.2.2
Cancel the common factor.
Step 14.3.16.2.3
Rewrite the expression.
Step 14.3.16.2.4
Divide by .
Step 14.3.17
Add and .
Step 14.3.18
Multiply by .
Step 14.3.19
Add and .
Step 15
The exact value of is .
Step 16
Simplify.
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Step 16.1
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 16.2
The exact value of is .
Step 16.3
Multiply by .
Step 16.4
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 16.5
The exact value of is .
Step 16.6
Multiply by .
Step 16.7
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 16.8
The exact value of is .
Step 16.9
Multiply by .
Step 16.10
Add and .
Step 16.11
Multiply .
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Step 16.11.1
Multiply by .
Step 16.11.2
Multiply by .
Step 16.12
Add and .
Step 16.13
Add and .
Step 17
The result can be shown in multiple forms.
Exact Form:
Decimal Form: