Definite Integration
Integration of rational functions
Grade Class 12
Question:
$\int \frac{x^2(1-\ln x)}{\ln^4 x - x^4} dx$ equals
(A) \frac{1}{2} \ln\left|\frac{x}{\ln x}\right| - \frac{1}{4} \ln|\ln^2 x - x^2| + C
(B) \frac{1}{4} \ln\left|\frac{\ln x - x}{\ln x + x}\right| - \frac{1}{2} \tan^{-1}\left(\frac{\ln x}{x}\right) + C
(C) \frac{1}{4} \ln\left|\frac{\ln x + x}{\ln x - x}\right| + \frac{1}{2} \tan^{-1}\left(\frac{\ln x}{x}\right) + C
(D) \frac{1}{4} \ln\left|\frac{\ln x - x}{\ln x + x}\right| + \tan^{-1}\left(\frac{\ln x}{x}\right) + C
Step-by-Step Solution
Key Concept: This is a complex integration problem likely requiring substitution or algebraic manipulation.
The integral requires careful substitution and partial fraction decomposition.
Correct Answer: B