\( \int\displaystyle\frac{dx}{x(x^{10}+1)} \) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{10}\log\left|\displaystyle\frac{x^{10}}{x^{10}+1}\right|+C \) , where C is an arbitrary constant
Multiplying numerator and denominator by \( x^9 \) brings in the derivative of \( x^{10} \):
\( \displaystyle \int\frac{dx}{x(x^{10}+1)}=\int\frac{x^9}{x^{10}(x^{10}+1)}\,dx \)
Put \( x^{10}=t \), so \( 10x^9\,dx=dt \):
\( \displaystyle =\frac{1}{10}\int\frac{dt}{t(t+1)} \)
\( \displaystyle =\frac{1}{10}\int\left(\frac{1}{t}-\frac{1}{t+1}\right)dt \)
\( \displaystyle =\frac{1}{10}\log\left|\frac{t}{t+1}\right|+C \)
\( \displaystyle =\frac{1}{10}\log\left|\frac{x^{10}}{x^{10}+1}\right|+C \)
Why the other options are incorrect
Option (1) \( \log\left|\displaystyle\frac{x^{10}}{x^{10}+1}\right|+C \) , where C is an arbitrary constant — Incorrect. The factor \( \frac{1}{10} \) from \( x^9\,dx=\frac{1}{10}\,dt \) is missing.
Option (2) \( \log\left|\displaystyle\frac{x^{10}+1}{x^{10}}\right|+C \) , where C is an arbitrary constant — Incorrect. The factor \( \frac{1}{10} \) is missing and the fraction is inverted, which changes the sign of the logarithm.
Option (3) \( \displaystyle\frac{1}{10}\log\left|\displaystyle\frac{x^{10}+1}{x^{10}}\right|+C \) , where C is an arbitrary constant — Incorrect. The fraction inside the logarithm is inverted; \( \log\left|\frac{t+1}{t}\right|=-\log\left|\frac{t}{t+1}\right| \), so the sign is wrong.
Tip: For \( \int\frac{dx}{x(x^n+1)} \) remember the standard result \( \frac{1}{n}\log\left|\frac{x^n}{x^n+1}\right|+C \); differentiating it checks both the factor and the order of the fraction.