Definite Integration
Integration by substitution
Grade Class 12
Question:
∫ \frac{1-x^7}{x(1+x^7)} dx \text{ equals -}
\ln x + \frac{2}{7} \ln(1+x^7) + c
\ln x - \frac{2}{7} \ln(1-x^7) + c
\ln x - \frac{2}{7} \ln(1+x^7) + c
\ln x + \frac{2}{7} \ln(1-x^7) + c
Step-by-Step Solution
Key Concept: The integral can be solved by splitting the integrand: (1-x^7)/(x(1+x^7)) = (1+x^7-2x^7)/(x(1+x^7)) = 1/x - 2x^6/(1+x^7). Integrating this gives ln|x| - 2/7 ln|1+x^7| + C.
We have $\int \frac{1-x^7}{x(1+x^7)} dx = \int \frac{1+x^7-2x^7}{x(1+x^7)} dx = \int \left( \frac{1}{x} - \frac{2x^6}{1+x^7} \right) dx$. Integrating term by term, we get $\ln|x| - \frac{2}{7} \ln|1+x^7| + c$.
Correct Answer: C