Calculus Examples

Evaluate the Integral 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
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
Let . Then , so . Rewrite using and .
Tap for more steps...
Step 6.1
Let . Find .
Tap for more steps...
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
Substitute and simplify.
Tap for more steps...
Step 10.1
Evaluate at and at .
Step 10.2
Evaluate at and at .
Step 10.3
Add and .
Step 11
Simplify.
Tap for more steps...
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
Simplify.
Tap for more steps...
Step 12.1
Simplify the numerator.
Tap for more steps...
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: