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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
Simplify each term.
Step 2.1.1.1
Apply the product rule to .
Step 2.1.1.2
Raise to the power of .
Step 2.1.1.3
Multiply by .
Step 2.1.2
Factor out of .
Step 2.1.3
Factor out of .
Step 2.1.4
Factor out of .
Step 2.1.5
Apply pythagorean identity.
Step 2.1.6
Rewrite as .
Step 2.1.7
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2
Simplify.
Step 2.2.1
Multiply by .
Step 2.2.2
Raise to the power of .
Step 2.2.3
Raise to the power of .
Step 2.2.4
Use the power rule to combine exponents.
Step 2.2.5
Add and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Use the half-angle formula to rewrite as .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Combine and .
Step 7
Split the single integral into multiple integrals.
Step 8
Apply the constant rule.
Step 9
Step 9.1
Let . Find .
Step 9.1.1
Differentiate .
Step 9.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 9.1.3
Differentiate using the Power Rule which states that is where .
Step 9.1.4
Multiply by .
Step 9.2
Substitute the lower limit in for in .
Step 9.3
Multiply by .
Step 9.4
Substitute the upper limit in for in .
Step 9.5
Cancel the common factor of .
Step 9.5.1
Cancel the common factor.
Step 9.5.2
Rewrite the expression.
Step 9.6
The values found for and will be used to evaluate the definite integral.
Step 9.7
Rewrite the problem using , , and the new limits of integration.
Step 10
Combine and .
Step 11
Since is constant with respect to , move out of the integral.
Step 12
The integral of with respect to is .
Step 13
Step 13.1
Evaluate at and at .
Step 13.2
Evaluate at and at .
Step 13.3
Add and .
Step 14
Step 14.1
The exact value of is .
Step 14.2
Multiply by .
Step 14.3
Add and .
Step 14.4
Combine and .
Step 15
Step 15.1
Combine the numerators over the common denominator.
Step 15.2
Simplify each term.
Step 15.2.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 15.2.2
The exact value of is .
Step 15.3
Add and .
Step 15.4
Multiply .
Step 15.4.1
Multiply by .
Step 15.4.2
Multiply by .
Step 16
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 17