Definite Integration
Substitution method
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: Use substitution method by multiplying numerator and denominator by x^6 to make the denominator (x^7(1+x^7)).
Let I = \int (1-x^7) / (x(1+x^7)) dx. Multiply numerator and denominator by x^6: I = \int (x^6(1-x^7)) / (x^7(1+x^7)) dx. Let t = x^7, then dt = 7x^6 dx. So, I = (1/7) \int (1-t) / (t(1+t)) dt. Using partial fractions: (1-t)/(t(1+t)) = 1/t - 2/(1+t). Thus, I = (1/7) [ln|t| - 2ln|1+t|] + c = (1/7) [ln|x^7| - 2ln|1+x^7|] + c = ln|x| - (2/7)ln|1+x^7| + c.
Correct Answer: C