Integral Evaluation: A Comprehensive Guide to the Integral of 1/{cos(x)cos(x1) sin(x)sin(x1)}

Understanding complex integrals involving trigonometric functions can be a challenging yet rewarding task. In this article, we aim to demystify the integral of

displaystyle int frac{1}{cos(x) cos(x_1) sin(x) sin(x_1)} dx

The goal is to evaluate this integral and explore the myriad of methods and substitutions that can be employed to solve it. We will discuss a trigonometric identity simplification and a series of substitution techniques to find the solution.

Simplifying the Integral

Starting with the given integral, we observe that the denominator can be simplified using trigonometric identities. Specifically, we use the angle addition formula for cosine:

cos(x) cos(x_1) sin(x) sin(x_1) cos(x - x_1)

This simplifies our integral to:

displaystyle int frac{1}{cos(x - x_1)} dx

Substitution 1: Using Trigonometric Substitution

To further simplify, let's substitute:

text{Let } t tan left( frac{x}{2} right)

Then, using the half-angle formulas:

cos(x) frac{1 - t^2}{1 t^2} quad sin(x) frac{2t}{1 t^2} quad dx frac{2}{1 t^2} dt

Substituting these into our integral, we get:

displaystyle int frac{1}{1 - cos x cos x_1 - sin x sin x_1} dx int frac{1}{cos(x - x_1)} dx int frac{2}{1 t^2} cdot frac{1 t^2}{1 - t} dt

Further simplifying, we obtain:

int frac{1 t^2}{1 - t} dt int frac{1}{1 - t} dt

Evaluating the integral:

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

Alternative Approach: Using Trigonometric Identities and Substitutions

Another approach involves using a different substitution and simplifying the integrand:

displaystyle int frac{1}{cos^2 x cos x sin^2 x sin x} dx

Using the identity sin^2 x cos^2 x 1, we can rewrite the integral:

displaystyle int frac{1}{1 - cos x sin x} dx

Further simplifying, we get:

displaystyle int frac{1}{2 sin x cos x} dx frac{1}{2} int frac{sin x}{sin x cos x} dx - frac{1}{2} int frac{1}{sin 2x} dx

Which leads us to:

frac{1}{2} int sec x dx - frac{1}{2} int csc 2x dx

Evaluating both integrals:

frac{1}{2} ln |sec x tan x| C - frac{1}{2} ln |csc 2x - cot 2x| D

Or simply:

frac{1}{2} ln |sec x tan x - csc 2x| C

Conclusion

Both methods yield the same result, providing a robust understanding of how to evaluate the integral. Whether through trigonometric identities or clever substitutions, the key lies in carefully simplifying the integrand.

Keywords: integral, trigonometric substitution, logarithmic integral, trigonometric identities