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Calculus Examples
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Step 2.1
Let . Find .
Step 2.1.1
Differentiate .
Step 2.1.2
Differentiate.
Step 2.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3
Evaluate .
Step 2.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.2
Differentiate using the chain rule, which states that is where and .
Step 2.1.3.2.1
To apply the Chain Rule, set as .
Step 2.1.3.2.2
Differentiate using the Power Rule which states that is where .
Step 2.1.3.2.3
Replace all occurrences of with .
Step 2.1.3.3
The derivative of with respect to is .
Step 2.1.3.4
Multiply by .
Step 2.1.3.5
Multiply by .
Step 2.1.4
Simplify.
Step 2.1.4.1
Add and .
Step 2.1.4.2
Reorder and .
Step 2.1.4.3
Reorder and .
Step 2.1.4.4
Apply the sine double-angle identity.
Step 2.2
Rewrite the problem using and .
Step 3
By the Power Rule, the integral of with respect to is .
Step 4
Step 4.1
Simplify.
Step 4.1.1
Combine and .
Step 4.1.2
Move to the denominator using the negative exponent rule .
Step 4.2
Simplify.
Step 4.3
Simplify.
Step 4.3.1
Multiply by .
Step 4.3.2
Combine and .
Step 4.3.3
Cancel the common factor of and .
Step 4.3.3.1
Factor out of .
Step 4.3.3.2
Cancel the common factors.
Step 4.3.3.2.1
Factor out of .
Step 4.3.3.2.2
Cancel the common factor.
Step 4.3.3.2.3
Rewrite the expression.
Step 4.3.4
Move the negative in front of the fraction.
Step 5
Replace all occurrences of with .