# Evaluate integral of sin(x)^7cos(x)^5 with respect to x

Factor out .
Simplify with factoring out.
Factor out of .
Rewrite as exponentiation.
Using the Pythagorean Identity, rewrite as .
Let . Then , so . Rewrite using and .
Let . Find .
Differentiate .
The derivative of with respect to is .
Rewrite the problem using and .
Expand .
Rewrite as .
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Move .
Move parentheses.
Move .
Move .
Move parentheses.
Move .
Move .
Move parentheses.
Move parentheses.
Move .
Multiply by .
Multiply by .
Multiply by .
Factor out negative.
Use the power rule to combine exponents.
Multiply by .
Factor out negative.
Use the power rule to combine exponents.
Multiply by .
Multiply by .
Use the power rule to combine exponents.
Use the power rule to combine exponents.
Subtract from .
Reorder and .
Move .
Split the single integral into multiple integrals.
By the Power Rule, the integral of with respect to is .
Since is constant with respect to , move out of the integral.
By the Power Rule, the integral of with respect to is .
By the Power Rule, the integral of with respect to is .
Simplify.
Combine and .
Simplify.
Replace all occurrences of with .
Reorder terms.
Evaluate integral of sin(x)^7cos(x)^5 with respect to x