Simplifying and Solving the Integral ( intfrac{dx}{cos^2x-sin^2x} ) in Multiple Methodologies

In this article, we explore the integral ( intfrac{dx}{cos^2x-sin^2x} ). This integral can be simplified and solved using various trigonometric identities and techniques. We will delve into three different methods to solve this integral, ensuring a comprehensive understanding of the problem.

Method 1: Simplifying Using Trigonometric Identities

To begin, we recognize that ( cos^2x - sin^2x cos 2x ). Hence, we can rewrite the integral as follows:

( intfrac{dx}{cos^2x-sin^2x} intfrac{dx}{cos 2x} )

We then apply the substitution ( t tan x ), giving us ( sec^2 x dx dt ). Substituting these into the integral, we get:

( intfrac{sec^2 x dx}{1 - tan^2 x} intfrac{dt}{1 - t^2} )

We further simplify the integral using partial fractions:

( intfrac{dt}{1 - t^2} frac{1}{2} intleft(frac{1}{1-t} frac{1}{1 t}right)dt )

Integrating both terms:

( frac{1}{2}lnleft|frac{1-t}{1 t}right| C )

Substituting back ( t tan x ), we obtain:

( frac{1}{2}lnleft|frac{1-tan x}{1 tan x}right| C )

And further simplifying using the trigonometric identity for tangent at ( x - frac{pi}{4} )::

( frac{1}{2}ln|tan(x - frac{pi}{4})| C )

Method 2: Direct Integration with Substitution

Another approach involves a direct integration with a substitution. We rewrite the integral as:

( intfrac{dx}{cos^2x - sin^2x} intsec 2x dx )

Using the identity ( sec^2 x 1 tan^2 x ), we multiply the numerator and denominator by ( tan 2x ) and ( sec 2x ) to integrate:

( frac{1}{2}intfrac{2sec^2 2xsec 2xtan 2x}{sec 2xtan 2x}dx )

This simplifies to:

( frac{1}{2}int d(sec 2xtan 2x) )

And finally, integrating this expression:

( frac{1}{2}ln|sec 2xtan 2x| C )

Method 3: Using Trigonometric Substitution

In this method, we use the substitution ( t 2x ), leading to ( dx frac{dt}{2} ). Substituting into the integral, we get:

( I intfrac{1}{cos^2 2x - sin^2 2x}frac{dt}{2} frac{1}{2}intsec 2x dx )

Using the identity ( sec 2x frac{1}{cos 2x} ), we can integrate:

( frac{1}{2}log|sec 2xtan 2x| C )

Thus, we obtain the final result in a compact form:

( boxed{frac{1}{2}log|sec 2xtan 2x| C} )

Conclusion

This article has explored three distinct methods to solve the integral ( intfrac{dx}{cos^2x-sin^2x} ). Each method leverages different trigonometric identities and integration techniques, providing a comprehensive view of the problem. The final result is:

( boxed{frac{1}{2}ln|tan(x-frac{pi}{4})|, frac{1}{2}ln|sec 2xtan 2x|, frac{1}{2}log|sec 2xtan 2x|} )