Evaluating Complex Integrals: Techniques and Step-by-Step Approach

Complex integrals often require a combination of algebraic manipulation, trigonometric identities, and integration techniques. In this article, we present a detailed step-by-step approach to evaluate the integral:

[int left frac{cos^2 x}{1 tan x} right dx]

Step 1: Simplify the Integral Expression

First, let's simplify the given integral:

[int left frac{cos^2 x}{1 tan x} right dx int left frac{cos^2 x}{1 frac{sin x}{cos x}} right dx int left frac{cos^3 x}{cos x sin x} right dx]

Step 2: Further Simplification

To simplify further, we can use the double-angle identities and algebraic manipulation:

[int left frac{cos^3 x}{cos x sin x} right dx int frac{cos x (cos^2 x - sin^2 x)}{cos x sin x} dx int frac{cos x (1 - 2sin^2 x)}{cos x sin x} dx]

Step 3: Use Trigonometric Substitution

Let's use the substitution:

[tan x t Rightarrow frac{dx}{dt} frac{1}{1 t^2}, quad dx frac{dt}{1 t^2}]

Therefore, the integral transforms to:

[I int frac{cos^2 x}{1 t} cdot frac{dt}{1 t^2}]

Since (cos^2 x frac{1}{1 tan^2 x} frac{1}{1 t^2}), the integral becomes:

[I int frac{1}{1 t} dt]

This integral can be easily evaluated:

[I ln|1 t| C ln|1 tan x| C]

However, the article suggests another approach that leads to a more complex but satisfactory result. Let's follow the detailed approach as mentioned in the reference.

Step 4: Breakdown of the Integral

The article further suggests breaking down the integral into simpler parts:

[I int frac{cos^2 x}{1 tan x} dx int left( frac{cos x - sin x}{4 sin x cos x} frac{2 sin x 2 cos x}{4 sin x cos x} - frac{sin x cos 2x cos x sin 2x}{4 sin x cos x} right) dx]

Step 5: Simplify Each Term

Breaking down each term:

1. (int frac{cos x - sin x}{4 sin x cos x} dx)

2. (int frac{2 (sin x cos x)}{4 sin x cos x} dx)

3. (int frac{sin x cos 2x cos x sin 2x}{4 sin x cos x} dx)

This simplifies to:

[I frac{1}{4} ln |sin x cos x| frac{1}{2} x - frac{1}{8} sin 2x cos 2x C]

Therefore, the final form of the integral is:

[int frac{cos^2 x}{1 tan x} dx frac{1}{8} [4x sin 2x cos 2x 2 ln |sin x cos x|] C]

Conclusion

This step-by-step approach provides a methodical and detailed solution to the problem of evaluating a complex integral. By breaking down the problem into simpler components and using appropriate substitutions and algebraic manipulations, we can effectively solve otherwise challenging integrals.