Calculus Examples

Evaluate the Integral integral from 0 to pi/2 of cos(x)^2sin(x) with respect to x
Step 1
Let . Then , so . Rewrite using and .
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Step 1.1
Let . Find .
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Step 1.1.1
Differentiate .
Step 1.1.2
The derivative of with respect to is .
Step 1.2
Substitute the lower limit in for in .
Step 1.3
The exact value of is .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
The exact value of is .
Step 1.6
The values found for and will be used to evaluate the definite integral.
Step 1.7
Rewrite the problem using , , and the new limits of integration.
Step 2
Since is constant with respect to , move out of the integral.
Step 3
By the Power Rule, the integral of with respect to is .
Step 4
Combine and .
Step 5
Substitute and simplify.
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Step 5.1
Evaluate at and at .
Step 5.2
Simplify.
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Step 5.2.1
Raising to any positive power yields .
Step 5.2.2
Cancel the common factor of and .
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Step 5.2.2.1
Factor out of .
Step 5.2.2.2
Cancel the common factors.
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Step 5.2.2.2.1
Factor out of .
Step 5.2.2.2.2
Cancel the common factor.
Step 5.2.2.2.3
Rewrite the expression.
Step 5.2.2.2.4
Divide by .
Step 5.2.3
One to any power is one.
Step 5.2.4
Subtract from .
Step 5.2.5
Multiply by .
Step 5.2.6
Multiply by .
Step 6
The result can be shown in multiple forms.
Exact Form:
Decimal Form: