Calculus Examples

Evaluate the Integral integral from 0 to pi/2 of cos(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
Split the single integral into multiple integrals.
Step 4
Apply the constant rule.
Step 5
Let . Then , so . Rewrite using and .
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Step 5.1
Let . Find .
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Step 5.1.1
Differentiate .
Step 5.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3
Differentiate using the Power Rule which states that is where .
Step 5.1.4
Multiply by .
Step 5.2
Substitute the lower limit in for in .
Step 5.3
Multiply by .
Step 5.4
Substitute the upper limit in for in .
Step 5.5
Cancel the common factor of .
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Step 5.5.1
Cancel the common factor.
Step 5.5.2
Rewrite the expression.
Step 5.6
The values found for and will be used to evaluate the definite integral.
Step 5.7
Rewrite the problem using , , and the new limits of integration.
Step 6
Combine and .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
The integral of with respect to is .
Step 9
Substitute and simplify.
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Step 9.1
Evaluate at and at .
Step 9.2
Evaluate at and at .
Step 9.3
Add and .
Step 10
Simplify.
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Step 10.1
The exact value of is .
Step 10.2
Multiply by .
Step 10.3
Add and .
Step 10.4
Combine and .
Step 11
Simplify.
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Step 11.1
Combine the numerators over the common denominator.
Step 11.2
Simplify each term.
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Step 11.2.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 11.2.2
The exact value of is .
Step 11.3
Add and .
Step 11.4
Multiply .
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Step 11.4.1
Multiply by .
Step 11.4.2
Multiply by .
Step 12
The result can be shown in multiple forms.
Exact Form:
Decimal Form: