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Calculus Examples
Step 1
Use the half-angle formula to rewrite as .
Step 2
Since is constant with respect to , move out of the integral.
Step 3
Split the single integral into multiple integrals.
Step 4
Apply the constant rule.
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Step 6.1
Let . Find .
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
Substitute the lower limit in for in .
Step 6.3
Multiply by .
Step 6.4
Substitute the upper limit in for in .
Step 6.5
The values found for and will be used to evaluate the definite integral.
Step 6.6
Rewrite the problem using , , and the new limits of integration.
Step 7
Combine and .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
The integral of with respect to is .
Step 10
Step 10.1
Evaluate at and at .
Step 10.2
Evaluate at and at .
Step 10.3
Add and .
Step 11
Step 11.1
The exact value of is .
Step 11.2
Multiply by .
Step 11.3
Add and .
Step 11.4
Combine and .
Step 12
Step 12.1
Simplify the numerator.
Step 12.1.1
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 12.1.2
The exact value of is .
Step 12.2
Divide by .
Step 12.3
Multiply by .
Step 12.4
Add and .
Step 12.5
Combine and .
Step 13
The result can be shown in multiple forms.
Exact Form:
Decimal Form: