Indefinite Integration
Integration Using Algebraic Manipulation
IIT-JEE 1996
Grade 12
Question:
$\int \frac{dx}{x(x^4 + 1)}$ equals:
(1) $\frac{1}{4} \ln\left(\frac{x^4}{x^4 + 1}\right) + C$
(2) $\frac{1}{4} \ln\left(\frac{x^4 + 1}{x^4}\right) + C$
(3) $\ln\left(\frac{x^4}{x^4 + 1}\right) + C$
(4) $\frac{1}{4} \ln(x^4 + 1) + C$
Step-by-Step Solution
Key Concept: Use the algebraic split $\frac{1}{x(x^4 + 1)} = \frac{1}{x} - \frac{x^3}{x^4 + 1}$ before integrating term by term.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)