Factor out .

Factor out of .

Rewrite as exponentiation.

Using the Pythagorean Identity, rewrite as .

Let . Find .

Differentiate .

The derivative of with respect to is .

Rewrite the problem using and .

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.

Add and .

Multiply by .

Factor out negative.

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Use the power rule to combine exponents.

Add and .

Use the power rule to combine exponents.

Add and .

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 .

Combine and .

Simplify.

Replace all occurrences of with .

Reorder terms.

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