Calculus Examples

Evaluate the Integral integral from 0 to pi/2 of sin(2x) 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
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3
Differentiate using the Power Rule which states that is where .
Step 1.1.4
Multiply by .
Step 1.2
Substitute the lower limit in for in .
Step 1.3
Multiply by .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Cancel the common factor of .
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Step 1.5.1
Cancel the common factor.
Step 1.5.2
Rewrite the expression.
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
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
The integral of with respect to is .
Step 5
Evaluate at and at .
Step 6
The exact value of is .
Step 7
Simplify.
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Step 7.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 7.2
The exact value of is .
Step 7.3
Multiply by .
Step 7.4
Multiply by .
Step 7.5
Add and .
Step 7.6
Cancel the common factor of .
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Step 7.6.1
Cancel the common factor.
Step 7.6.2
Rewrite the expression.