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Calculus Examples
Step 1
Split the single integral into multiple integrals.
Step 2
Apply the constant rule.
Step 3
Let , where . Then . Note that since , is positive.
Step 4
Step 4.1
Simplify .
Step 4.1.1
Simplify each term.
Step 4.1.1.1
Apply the product rule to .
Step 4.1.1.2
Raise to the power of .
Step 4.1.1.3
Multiply by .
Step 4.1.2
Factor out of .
Step 4.1.3
Factor out of .
Step 4.1.4
Factor out of .
Step 4.1.5
Apply pythagorean identity.
Step 4.1.6
Rewrite as .
Step 4.1.7
Pull terms out from under the radical, assuming positive real numbers.
Step 4.2
Simplify.
Step 4.2.1
Multiply by .
Step 4.2.2
Raise to the power of .
Step 4.2.3
Raise to the power of .
Step 4.2.4
Use the power rule to combine exponents.
Step 4.2.5
Add and .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Use the half-angle formula to rewrite as .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Combine and .
Step 9
Split the single integral into multiple integrals.
Step 10
Apply the constant rule.
Step 11
Step 11.1
Let . Find .
Step 11.1.1
Differentiate .
Step 11.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 11.1.3
Differentiate using the Power Rule which states that is where .
Step 11.1.4
Multiply by .
Step 11.2
Substitute the lower limit in for in .
Step 11.3
Cancel the common factor of .
Step 11.3.1
Move the leading negative in into the numerator.
Step 11.3.2
Cancel the common factor.
Step 11.3.3
Rewrite the expression.
Step 11.4
Substitute the upper limit in for in .
Step 11.5
Multiply by .
Step 11.6
The values found for and will be used to evaluate the definite integral.
Step 11.7
Rewrite the problem using , , and the new limits of integration.
Step 12
Combine and .
Step 13
Since is constant with respect to , move out of the integral.
Step 14
The integral of with respect to is .
Step 15
Step 15.1
Evaluate at and at .
Step 15.2
Evaluate at and at .
Step 15.3
Evaluate at and at .
Step 15.4
Simplify.
Step 15.4.1
Add and .
Step 15.4.2
Add and .
Step 16
Step 16.1
The exact value of is .
Step 16.2
Subtract from .
Step 16.3
Combine and .
Step 17
Step 17.1
Simplify the numerator.
Step 17.1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 17.1.2
The exact value of is .
Step 17.2
Divide by .
Step 17.3
Multiply by .
Step 17.4
Add and .
Step 17.5
Multiply .
Step 17.5.1
Multiply by .
Step 17.5.2
Multiply by .
Step 18
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 19