Evaluating the Integral of 1 / (sin4x cos4x sin2x cos2x): Detailed Steps and Techniques

In this article, we will explore a detailed approach to evaluate the integral of the function 1 / (sin4x cos4x sin2x cos2x) using trigonometric identities and substitution techniques. This integral is a challenging problem that requires a systematic method to simplify and solve it. We'll break down the process step-by-step and explain each step to provide a comprehensive understanding.

Step-by-Step Evaluation of the Integral

Let's begin by examining the integral:

( I int frac{1}{sin^4 x cos^4 x sin^2 x cos^2 x} , dx )

First, we can simplify the integrand using the identity ( sin^2 x cos^2 x frac{1}{4} sin^2 2x ). Let's apply this identity:

( I int frac{1}{left(sin^2 x cos^2 xright)^2 - sin^2 x cos^2 x} , dx )

Substituting the identity:

( I int frac{1}{left[sin^2 x cos^2 xright]^2 - sin^2 x cos^2 x} , dx )

Simplify the denominator:

( I int frac{1}{left[1 - 2 sin^2 x cos^2 xright]} , dx )

Next, express ( sin^2 2x ) in terms of ( sin x cos x ):

( sin^2 2x 4 sin^2 x cos^2 x ) and thus,

( I int frac{1}{1 - frac{1}{4} sin^2 2x} , dx )

Now, let's substitute ( u sin 2x ) and ( frac{du}{dx} 2 cos 2x ), so ( dx frac{1}{2 cos 2x} , du ):

( I int frac{1}{1 - frac{u^2}{4}} cdot frac{1}{2 cos 2x} , du )

Since ( cos 2x sqrt{1 - sin^2 2x} ), we have:

( I int frac{1}{1 - frac{u^2}{4}} cdot frac{1}{2 sqrt{1 - u^2}} , du )

Simplify and further substitute ( u tan x ) and ( du sec^2 x , dx ):

( I int frac{1}{1 - frac{tan^2 x}{4}} cdot frac{sec^2 x}{2 sqrt{1 - tan^2 x}} , dx )

Using the identity ( sec^2 x 1 tan^2 x ), the integral becomes:

( I int frac{1 tan^2 x}{1 - frac{tan^2 x}{4} tan^2 x (1 - frac{1}{4})} cdot frac{1}{2 sqrt{1 - tan^2 x}} , dx )

After simplification, we get:

( I frac{2}{sqrt{3}} int frac{2sqrt{3} sec^2 x}{4sqrt{3} tan 2x} , dx )

The integral simplifies to:

( I frac{1}{sqrt{3}} tan^{-1} left(frac{sqrt{3} tan 2x}{2}right) C )

Thus, we have:

( I boxed{frac{1}{sqrt{3}} tan^{-1} left(frac{sqrt{3} tan 2x}{2}right) C} )

Conclusion

The integral ( int frac{1}{sin^4 x cos^4 x sin^2 x cos^2 x} , dx ) was successfully evaluated using trigonometric identities and substitution techniques. This process involved simplifying the integrand, transforming it into a more manageable form, and applying appropriate substitutions. Understanding these steps can help in solving similar complex integrals involving trigonometric functions.

Related Topics

This article covers several related topics that can be explored further:

Integral Evaluation: Techniques and methods for evaluating integrals. Trigonometric Substitution: Using trigonometric identities and substitutions to simplify integrals. Arctangent Integration: Techniques for integrating functions that can be expressed in terms of the arctangent function.