Evaluating Integrals Involving Logarithms and Trigonometric Functions

In this article, we will explore the intricacies of evaluating integrals involving logarithms and trigonometric functions. Specifically, we will delve into the evaluation of the integral of the natural logarithm of the cosine function over a specific interval.

Introduction

The integral in question is given by:

[ I int_0^{frac{pi}{4}} ln(cos x) , dx ]

Similarly, we define another integral:

[ J int_0^{frac{pi}{4}} ln(sin x) , dx ]

By evaluating these integrals and using some algebraic manipulation, we can derive their values.

Deriving the Integral Values

First, let's consider the product of the two integrals:

[ I cdot J int_0^{frac{pi}{4}} ln(cos x) int_0^{frac{pi}{4}} ln(sin x) , dx ]

This simplifies to:

[ I cdot J int_0^{frac{pi}{4}} [ln(cos x) cdot ln(sin x)] , dx ]

Using the identity ( cos x cdot sin x frac{1}{2} sin(2x) ), we have:

[ I cdot J int_0^{frac{pi}{4}} [ln(frac{1}{2}sin(2x)) - ln(2)] , dx ]

This can further be broken down:

[ I cdot J int_0^{frac{pi}{4}} ln(sin(2x)) , dx - ln(2) int_0^{frac{pi}{4}} , dx ]

By making the substitution (2x u implies dx frac{1}{2} du), we transform the integral:

[ I cdot J frac{1}{2} int_0^{frac{pi}{2}} ln(sin u) , du - frac{pi}{4} ln(2) ]

It is known that:

[ int_0^{frac{pi}{2}} ln(sin u) , du -frac{pi}{2} ln(2) ]

Thus:

[ I cdot J frac{1}{2} left(-frac{pi}{2} ln(2)right) - frac{pi}{4} ln(2) ]

Simplifying this, we get:

[ I cdot J -frac{pi}{4} ln(2) ]

Next, we consider the difference between the integrals:

[ I - J int_0^{frac{pi}{4}} ln(cos x) , dx - int_0^{frac{pi}{4}} ln(sin x) , dx ]

This simplifies to:

[ I - J int_0^{frac{pi}{4}} lnleft(frac{cos x}{sin x}right) , dx ]

Using the identity (cot x frac{cos x}{sin x}), we have:

[ I - J int_0^{frac{pi}{4}} ln(cot x) , dx G ]

Where (G) is the Catalan's constant.

Solving for the Integrals

Using the equations from above, we can solve for (I) and (J):

From (I cdot J -frac{pi}{4} ln(2)) and (I - J G), we solve for (I) and (J):

Assume (I J A) and (I - J B). Thus:

[ 2I A - B -frac{pi}{4} ln(2) - G ] [ I frac{G}{2} - frac{pi}{4} ln(2) ]

Similarly:

[ 2J A B -frac{pi}{4} ln(2) G ] [ J -frac{G}{2} - frac{pi}{4} ln(2) ]

Thus, the integral of the natural logarithm of the cosine function over the interval ([0, frac{pi}{4}]) is:

[ boxed{int_0^{frac{pi}{4}} ln(cos x) , dx frac{G}{2} - frac{pi}{4} ln(2) quad text{where } G text{ is Catalan's constant}} ]

Conclusion

In this article, we have discussed the steps to evaluate the given integral and derived its explicit value. Such evaluations are important in various fields, such as physics, engineering, and pure mathematics. The understanding of such integrals can help in solving complex problems and deepening the knowledge of mathematical analysis.