Calculus Examples

Evaluate the Integral pi integral from 0 to pi of sin(x)^2 with respect to x
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
Combine and .
Step 4
Split the single integral into multiple integrals.
Step 5
Apply the constant rule.
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Let . Then , so . Rewrite using and .
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Step 7.1
Let . Find .
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Step 7.1.1
Differentiate .
Step 7.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 7.1.3
Differentiate using the Power Rule which states that is where .
Step 7.1.4
Multiply by .
Step 7.2
Substitute the lower limit in for in .
Step 7.3
Multiply by .
Step 7.4
Substitute the upper limit in for in .
Step 7.5
The values found for and will be used to evaluate the definite integral.
Step 7.6
Rewrite the problem using , , and the new limits of integration.
Step 8
Combine and .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
The integral of with respect to is .
Step 11
Substitute and simplify.
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Step 11.1
Evaluate at and at .
Step 11.2
Evaluate at and at .
Step 11.3
Add and .
Step 12
Simplify.
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Step 12.1
The exact value of is .
Step 12.2
Multiply by .
Step 12.3
Add and .
Step 12.4
Combine and .
Step 13
Simplify.
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Step 13.1
Simplify the numerator.
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Step 13.1.1
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 13.1.2
The exact value of is .
Step 13.2
Divide by .
Step 13.3
Multiply by .
Step 13.4
Add and .
Step 13.5
Multiply .
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Step 13.5.1
Combine and .
Step 13.5.2
Raise to the power of .
Step 13.5.3
Raise to the power of .
Step 13.5.4
Use the power rule to combine exponents.
Step 13.5.5
Add and .
Step 14
The result can be shown in multiple forms.
Exact Form:
Decimal Form: