Evaluating the Integral of ln(tan(x)) from 0 to pi/4: A Calculus Challenge

The integral of ln(tan(x)) from 0 to pi/4 is a challenging problem in calculus, often encountered in advanced undergraduate or graduate courses. This integral is not only a fascinating problem in itself but also has connections to well-known constants in mathematics, such as Catalan's constant. In this article, we will explore various methods to evaluate this integral and understand the significance of the result.

Introduction to the Integral

Consider the definite integral:

[ int_{0}^{frac{pi}{4}} ln(tan x) , dx ]

This integral does not have a straightforward antiderivative, making it a non-trivial problem to solve. However, through a series of substitutions and advanced calculus techniques, we can evaluate this integral.

Substitution Techniques

One method to simplify the integral is to use the substitution x arctan u. This transforms the integral into:

[ int_{0}^{1} frac{ln u}{1 u^2} , du ]

This integral can be evaluated using a series expansion or other advanced techniques. Another approach is to use the geometric series expansion:

[ frac{1}{1-x} sum_{k0}^{infty} x^k ]

By substituting x -u^2, we can rewrite the integral as:

[ int_{0}^{1} frac{ln u}{1 u^2} , du int_{0}^{1} ln u sum_{k0}^{infty} (-1)^k u^{2k} , du ]

Integration by Parts and Catalan's Constant

To proceed, we use integration by parts, where we set:

u ln u dv sum_{k0}^{infty} (-1)^k u^{2k} du

By applying integration by parts, we get:

[ int_{0}^{1} ln u sum_{k0}^{infty} (-1)^k u^{2k} , du -sum_{k0}^{infty} frac{(-1)^k}{(2k 1)^2} ]

The series on the right-hand side is known as Catalan's constant, denoted by G. Therefore, we can write:

[ int_{0}^{frac{pi}{4}} ln(tan x) , dx -G ]

Conclusion

The integral of ln(tan(x)) from 0 to pi/4 is a powerful example of the interplay between advanced calculus techniques and well-known constants in mathematics. By using substitution and integration by parts, we can evaluate this integral and conclude that: