Evaluating the Integral of x / (sin x cos x cos x) from 0 to π/4: A Detailed Guide

Understanding and evaluating integrals accurately is essential for many mathematical and scientific applications. This article focuses on the integral ∫0π/4 x / (sin x cos x cos x) dx. By breaking down the integral and employing various integration techniques, we will evaluate this integral step by step. This approach not only solves the given problem but also serves as a powerful tool for general integral evaluation.

Step-by-Step Solution

We begin by manipulating the integral to make it more manageable:

∫0π/4 x / (sin x cos x cos x) dx ∫0π/4 (π/4 - x) / [sin(π/4 - x) cos(π/4 - x) cos(π/4 - x)] dx

Applying trigonometric identities, we simplify the integrand:

∫0π/4 (π/4 - x) / [sin(π/4 - x) cos(π/4 - x) cos(π/4 - x)] dx ∫0π/4 (π/4 - x) / [(1/√2 cos x - 1/√2 sin x) (1/√2 cos x - 1/√2 sin x) (1/√2 cos x - 1/√2 sin x)] dx

Further simplification yields:

∫0π/4 (π/4 - x) / (2/√2 cos x) dx ∫0π/4 (π/4 - x) / (sin x cos x cos x) dx

Next, we split the integral into two parts to make it easier to solve:

2∫0π/4 x / (sin x cos x cos x) dx (π/4) ∫0π/4 (1 / (sin x cos x cos x)) dx

Continuing, we further break down the integrand:

(π/4) ∫0π/4 (sin^2 x cos^2 x / (sin x cos x cos x)) dx (π/4) ∫0π/4 (sin x sin x cos x / (sin x cos x cos x)) dx - (π/4) ∫0π/4 (x / (sin x cos x cos x)) dx

Which simplifies to:

(π/4) ∫0π/4 (sin x / cos x) dx - (π/4) ∫0π/4 (cos x - sin x) / (sin x cos x) dx

Further simplification provides:

- (π/4) ∫0π/4 -sin x / cos x dx (π/4) ∫0π/4 d(sin x cos x) / (sin x cos x)

Finding the antiderivatives, we get:

- (π/4) [ln |cos x|]?π/4 (π/4) [ln |sin x cos x|]?π/4

Evaluating the limits, we obtain:

(() - (π/4) ln(1/√2) (π/4) ln(2/√2) (π/4) ln(2/1/√2) (π/4) ln 2)

Thus, the value of the integral is:

∫0π/4 x / (sin x cos x cos x) dx π/8 * ln 2

General Techniques for Integral Evaluation

1. Substitution: By using substitution, the original integral can be transformed into a more manageable form. In this case, we used the substitution x π/4 - u.

2. Trigonometric Identities: Applying trigonometric identities helps in simplifying the integrand. Here, we utilized the identities for sine and cosine products and sums.

3. Partial Fractions: While not explicitly used here, decomposition into partial fractions can be a useful technique for more complex integrals.

Conclusion

Evaluating integrals like ∫0π/4 x / (sin x cos x cos x) dx requires a systematic approach. By breaking down the integral into simpler parts and applying appropriate techniques, the problem becomes tractable. This guide provides a detailed step-by-step solution that not only evaluates the specific integral but also offers a broader understanding of integral evaluation techniques in calculus.

Keywords

integral evaluation, calculus, trigonometric functions, integration techniques