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Calculus Examples
Step 1
Let , where . Then . Note that since , is positive.
Step 2
Step 2.1
Simplify .
Step 2.1.1
Apply pythagorean identity.
Step 2.1.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2
Simplify.
Step 2.2.1
Raise to the power of .
Step 2.2.2
Raise to the power of .
Step 2.2.3
Use the power rule to combine exponents.
Step 2.2.4
Add and .
Step 3
Use the half-angle formula to rewrite as .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Split the single integral into multiple integrals.
Step 6
Apply the constant rule.
Step 7
Step 7.1
Let . Find .
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
Cancel the common factor of .
Step 7.5.1
Cancel the common factor.
Step 7.5.2
Rewrite the expression.
Step 7.6
The values found for and will be used to evaluate the definite integral.
Step 7.7
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
Step 11.1
Evaluate at and at .
Step 11.2
Evaluate at and at .
Step 11.3
Add and .
Step 12
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
Step 13.1
Combine the numerators over the common denominator.
Step 13.2
Simplify each term.
Step 13.2.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 13.2.2
The exact value of is .
Step 13.3
Add and .
Step 13.4
Multiply .
Step 13.4.1
Multiply by .
Step 13.4.2
Multiply by .
Step 14
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 15