To solve the integral \(\int \frac{\tan x}{\log (\cos x)} \, dx\), we need to analyze the given integrand thoroughly.
1. Identify the Integral and Substitutions:
We start with the integral:
[tex]\[
\int \frac{\tan x}{\log (\cos x)} \, dx
\][/tex]
2. Break Down the Integrand:
The integrand is \(\frac{\tan x}{\log (\cos x)}\), where:
- \(\tan x\) is the tangent of \(x\), which is \(\frac{\sin x}{\cos x}\)
- \(\log (\cos x)\) is the natural logarithm of \(\cos x\)
3. Simplification of the Integrand (Conceptual Step):
- The tangent function, \(\tan x = \frac{\sin x}{\cos x}\)
- Hence, \(\frac{\tan x}{\log (\cos x)} = \frac{\sin x}{\cos x \cdot \log (\cos x)}\)
4. Integration Strategy:
The complexity of the logarithmic function combined with the trigonometric function suggests that this integral is non-trivial and requires advanced techniques or possibly special functions for exact integration.
5. Representation of the Integral:
Given the above observations and the non-trivial nature of the functions involved, the integral can be expressed symbolically.
6. Result:
Hence, the integral \(\int \frac{\tan x}{\log (\cos x)} \, dx\) is represented as:
[tex]\[
\int \frac{\tan x}{\log (\cos x)} \, dx
\][/tex]
Thus, the integral \(\int \frac{\tan x}{\log (\cos x)} \, dx\) does not simplify further into elementary functions, and is best expressed in its integral form:
[tex]\[
\int \frac{\tan x}{\log (\cos x)} \, dx
\][/tex]
This is the most accurate representation of the integral given the complexity of the integrand functions involved.
