Definite Integration
Integration
Grade Class 12
Question:
∫ \frac{1-x^7}{x(1+x^7)} dx 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 into 1/x - 2x^6/(1+x^7) and then using substitution u = 1+x^7.
We have I = \int (1-x^7)/(x(1+x^7)) dx. Note that (1-x^7)/(x(1+x^7)) = (1+x^7-2x^7)/(x(1+x^7)) = 1/x - 2x^6/(1+x^7). Integrating term by term, we get \int 1/x dx - \int 2x^6/(1+x^7) dx = ln|x| - 2/7 ln|1+x^7| + c.
Correct Answer: 3