Understanding and Solving the Integral of cos^5(2x) / sin^3(2x)

The integral of trigonometric functions can be solved using various techniques such as substitution, trigonometric identities, and algebraic manipulation. In this article, we will explore the detailed steps to solve the integral of cos^5(2x) / sin^3(2x).

Introduction to the Problem

We start with the indefinite integral:

Symbols and Notations

Let I represent the given indefinite integral:

I ∫(cos^5(2x) / sin^3(2x)) dx

Substitution Method

The first step is to perform a substitution to simplify the integrand:

Substitution: u 2x

This substitution implies that dx (1/2) du, so the integral can be rewritten as:

∫(cos^5(2x) / sin^3(2x)) dx (1/2) ∫(cos^5(u) / sin^3(u)) du

Breaking Down the Numerator

Next, we factor one cos(u) out of the numerator and express cos^4(u) in terms of sin(u) using the Pythagorean identity:

cos^2(u) 1 - sin^2(u)

Thus, we can rewrite the numerator as:

cos^5(u) cos(u) * (1 - sin^2(u))^2

Substitute this back into the integral:

(1/2) ∫(cos(u) * (1 - sin^2(u))^2 / sin^3(u)) du

Further expanding and simplifying:

(1/2) ∫((1 - 2sin^2(u) sin^4(u)) / sin^3(u)) cos(u) du

Substitution: t sin(u)

Set the substitution t sin(u), which implies dt cos(u) du. Substitute this into the integral:

(1/2) ∫((1 - 2t^2 t^4) / t^3) dt

Split the fraction and simplify common terms:

(1/2) ∫(1/t^3 - 2/t - t) dt

Integration Steps

Now, we can integrate each term separately:

(1/2) [-1/(2t^2) - 2ln|t| - t^2 / 2]

Substitute t sin(u) back into the integration:

(1/2) [-1/(2sin^2(u)) - 2ln|sin(u)| - sin^2(u) / 2]

Simplify further and substitute back u 2x and t sin(2x):

-1/(4sin^2(2x)) - 2ln|sin(2x)| - sin^2(2x) / 4 C

Using the fact that 1/sin(2x) csc(2x), we can express the final answer more neatly:

I (sin^2(2x) - csc^2(2x)) / 4 - 2ln|sin(2x)| C

Conclusion

In conclusion, the integral of cos^5(2x) / sin^3(2x) dx is: