Calculus Examples

Evaluate the Integral integral from pi/3 to 2pi of 3cos(x)^2sin(x) with respect to x
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Let . Then , so . Rewrite using and .
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Step 2.1
Let . Find .
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Step 2.1.1
Differentiate .
Step 2.1.2
The derivative of with respect to is .
Step 2.2
Substitute the lower limit in for in .
Step 2.3
The exact value of is .
Step 2.4
Substitute the upper limit in for in .
Step 2.5
Simplify.
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Step 2.5.1
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 2.5.2
The exact value of is .
Step 2.6
The values found for and will be used to evaluate the definite integral.
Step 2.7
Rewrite the problem using , , and the new limits of integration.
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Multiply by .
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Combine and .
Step 7
Substitute and simplify.
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Step 7.1
Evaluate at and at .
Step 7.2
One to any power is one.
Step 8
Simplify.
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Step 8.1
Combine the numerators over the common denominator.
Step 8.2
Simplify each term.
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Step 8.2.1
Apply the product rule to .
Step 8.2.2
One to any power is one.
Step 8.2.3
Raise to the power of .
Step 8.3
Write as a fraction with a common denominator.
Step 8.4
Combine the numerators over the common denominator.
Step 8.5
Subtract from .
Step 8.6
Cancel the common factor of .
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Step 8.6.1
Factor out of .
Step 8.6.2
Cancel the common factor.
Step 8.6.3
Rewrite the expression.
Step 9
The result can be shown in multiple forms.
Exact Form:
Decimal Form: