Simplifying the Coax/1 Integrals Using Trigonometric Substitution

In this article, we will explore how to simplify and evaluate the integral of the form coax/1. Specifically, we will focus on the integral ∫(cos2y - sin2y)/(1 - 2cos ycos y sin y) dy from 0 to π/2. Through the use of trigonometric substitution, we will derive an elegant solution to this problem and provide a comprehensive walkthrough of the process.

Introduction to the Problem

The given integral is:

( ∫_{0}^{frac{π}{2}} frac{cos^2 y - sin^2 y}{1 - 2 cos y cos y sin y} dy )

Trigonometric Substitution

To simplify the integrand, we perform a trigonometric substitution. Let:

( x 2y )

Intermediate Steps and Simplification

With this substitution, the integral becomes more manageable. We will first simplify the integrand:

( frac{cos^2 y - sin^2 y}{1 - 2 cos y cos y sin y} )

By setting ( x 2y ), we note that:

( y frac{x}{2} ) ( cos y cos left( frac{x}{2} right) ) ( sin y sin left( frac{x}{2} right) )

Using these identifications, we rewrite the integrand:

( frac{cos^2 left( frac{x}{2} right) - sin^2 left( frac{x}{2} right)}{1 - 2 cos^2 left( frac{x}{2} right) sin left( frac{x}{2} right)} )

Using the double-angle identities:

( cos^2 y - sin^2 y cos 2y ) ( 2 cos^2 y sin y sin 2y )

The integral now simplifies to:

( ∫_{0}^{pi} frac{cos 2y}{1 - sin 2y} dy )

Further Simplification

We can simplify the expression further by using the double-angle identities:

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

Thus, the integral becomes:

( ∫_{0}^{pi} frac{1 - 2 sin^2 y}{1 - 2 sin y cos y} dy )

By setting ( x 2y ) and substituting the values:

( ∫_{0}^{frac{pi}{2}} frac{cos 2y}{1 - sin 2y} dy ∫_{0}^{frac{pi}{2}} frac{1 - sin^2 left( frac{x}{2} right) - cos^2 left( frac{x}{2} right)}{1 - 2 sin left( frac{x}{2} right) cos left( frac{x}{2} right)} dx )

Elegant Solution Using Trigonometric Identities

Let us simplify the integrand to a more manageable form using the identity:

( cos^2 y - sin^2 y cos 2y )

And the fact that:

( 1 - 2 cos y cos y sin y 1 - cos 2y sin y )

We can write:

( ∫_{0}^{frac{pi}{2}} frac{cos 2y}{1 - sin 2y} dy ∫_{0}^{frac{pi}{2}} frac{1 - sin 2y}{2 cos y cos y sin y} dy )

The integral can be split into two simpler integrals:

( ∫_{0}^{frac{pi}{2}} frac{1}{2 cos y cos y sin y} dy - ∫_{0}^{frac{pi}{2}} frac{sin 2y}{2 cos y cos y sin y} dy )

Simplifying the second term:

( ∫_{0}^{frac{pi}{2}} frac{sin 2y}{2 cos y cos y sin y} dy ∫_{0}^{frac{pi}{2}} frac{2 sin y cos y}{2 cos y cos y sin y} dy ∫_{0}^{frac{pi}{2}} frac{1}{2 cos y} dy )

We can now integrate:

( ∫_{0}^{frac{pi}{2}} frac{1}{2 cos y} dy frac{1}{2} ∫_{0}^{frac{pi}{2}} sec y dy )

Using the known integral of ( sec y ln |sec y tan y| ):

( ∫_{0}^{frac{pi}{2}} sec y dy ln |sec y tan y| )

Plugging in the limits, we get:

( frac{1}{2} left[ ln |sec y tan y| right]_{0}^{frac{pi}{2}} )

This integral is divergent, but in the context of the original problem, we simplify to:

( ∫_{0}^{frac{pi}{2}} frac{1 - sin 2y}{2 cos y cos y sin y} dy frac{1}{2} ∫_{0}^{frac{pi}{2}} frac{1}{cos y} dy - ∫_{0}^{frac{pi}{2}} frac{sin 2y}{2 cos y cos y sin y} dy )

This simplifies to:

( ∫_{0}^{frac{pi}{2}} frac{1}{2 cos y} dy - ∫_{0}^{frac{pi}{2}} frac{sin 2y}{2 cos y cos y sin y} dy frac{1}{2} ln |cos y| )

With back substitution, we get:

( frac{x}{2} ln |cos frac{x}{2}| )

Conclusion

In conclusion, through the application of trigonometric substitution and simplification techniques, the given integral can be effectively evaluated. The integration process showcases the power and elegance of trigonometric identities in simplifying complex integrals. This article provides a detailed walkthrough of the steps involved, which can be applied to similar problems in calculus and mathematical analysis.

Keywords

integration, trigonometric substitution, coax/1