# Evaluate integral from 0 to pi/3 of tan(x)^5sec(x)^4 with respect to x

Simplify the expression.
Rewrite as plus
Rewrite as .
Using the Pythagorean Identity, rewrite as .
Let . Then , so . Rewrite using and .
Let . Find .
Differentiate .
The derivative of with respect to is .
Substitute the lower limit in for in .
The exact value of is .
Substitute the upper limit in for in .
The exact value of is .
The values found for and will be used to evaluate the definite integral.
Rewrite the problem using , , and the new limits of integration.
Expand .
Apply the distributive property.
Reorder and .
Multiply by .
Use the power rule to combine exponents.
Reorder and .
Split the single integral into multiple integrals.
By the Power Rule, the integral of with respect to is .
By the Power Rule, the integral of with respect to is .
Combine and .
Substitute and simplify.
Evaluate at and at .
Simplify.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Combine and .
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Combine and .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Raising to any positive power yields .
Multiply by .
Raising to any positive power yields .
Multiply by .