Integration of dx/sin3x cos3x: A Comprehensive Guide

In this article, we will explore the integral dx/sin3x cos3x. This integral involves a combination of trigonometric functions and is a classic example of an integral that requires careful manipulation and knowledge of trigonometric identities and substitution techniques.

Introduction to the Problem

The integral we aim to solve is:

[ I int frac{dx}{sin^3 x cos^3 x} ]

Method 1: Substitution with Trigonometric Identities

We start by using the identity ( sin^2 x cos^2 x 1 ). Rearrange the integrand to factor out ( sin x cos x ) and simplify:

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

Further simplification gives:

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

Recognizing that ( 2 - sin^2 x - cos^2 x 1 - sin^2 x - cos^2 x 1 - 1 0 ), we can rewrite the integral as:

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

Let ( t sin x - cos x ). Then, ( sin x cos x dx dt ). Substituting yields:

[ I int frac{2 dt}{sin x cos x (2 - t^2)} ]

This can be further simplified as:

[ I int frac{2 dt}{1 - t^2} - int frac{2 dt}{2 - t^2} ]

These integrals can be evaluated using standard logarithmic and arctangent forms:

[ I ln left| frac{1-t}{1 t} right| - frac{1}{sqrt{2}} ln left| frac{sqrt{2}-t}{sqrt{2} t} right| C ]

Substituting back ( t sin x - cos x ), we obtain the final form:

[ I ln left| frac{1 - sin x cos x}{1 - sin x cos x} right| - frac{1}{sqrt{2}} ln left| frac{sqrt{2} - sin x cos x}{sqrt{2} sin x cos x} right| C ]

Method 2: Using Trigonometric Identities and Substitution

Alternatively, we can express the integrand using the product-to-sum identities and trigonometric substitutions:

Let ( I int frac{1}{sin^3 x cos^3 x} dx ). We use the identity:

[ sin^3 x cos^3 x left( sin x cos x right) left( sin^2 x - sin x cos x cos^2 x right) left( sin x cos x right) left( 3 - left( sin x cos x right)^2 right) ]

Then, the integrand can be rewritten as:

[ frac{1}{sin^3 x cos^3 x} frac{1}{3 left( sin x cos x right) left( sqrt{3} - left( sin x cos x right) right) left( sqrt{3} left( sin x cos x right) right)} ]

This can be split into partial fractions:

[ frac{1}{3} left[ frac{2}{sin x cos x} - frac{1}{sqrt{3} - sin x cos x} - frac{1}{sqrt{3} sin x cos x} right] ]

Each term can be integrated separately. The first term can be simplified using:

[ frac{2}{sin x cos x} frac{sqrt{2}}{sin x cos left( frac{pi}{4} right)} sqrt{2} csc left( x frac{pi}{4} right) ]

The remaining integrals can be solved using standard forms:

[ I -sqrt{2} log left[ cot left( x frac{pi}{4} right) csc left( x frac{pi}{4} right) right] - frac{2sqrt{2}}{3sqrt{5}} arctan left[ sqrt{frac{6}{5}} tan left( x frac{pi}{8} right) frac{1}{sqrt{5}} right] frac{2sqrt{2}}{3sqrt{5}} arctan left[ sqrt{frac{6}{5}} tan left( x frac{pi}{8} right) - frac{1}{sqrt{5}} right] C ]

Conclusion

This integral demonstrates the power of trigonometric identities and substitution techniques in integration. By carefully manipulating the integrand and using appropriate substitutions, we can find the solution efficiently. Understanding these techniques is crucial for solving more complex integrals in calculus.

Keywords: integral dx, integration techniques, trigonometric integrals