Value of \( \int_{1}^{4}\displaystyle\frac{\log_{e}x}{x^{2}}\,dx \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{1}{4}(3-\log_{e}4) \)
Integrating by parts, with \( \log_{e}x \) as the first function and \( \frac{1}{x^{2}} \) as the second:
\( \displaystyle \int\frac{\log_{e}x}{x^{2}}\,dx=\log_{e}x\cdot\left(-\frac{1}{x}\right)-\int\frac{1}{x}\cdot\left(-\frac{1}{x}\right)dx \)
\( \displaystyle =-\frac{\log_{e}x}{x}-\frac{1}{x} \)
So
\( \displaystyle \int_{1}^{4}\frac{\log_{e}x}{x^{2}}\,dx=\left[-\frac{\log_{e}x+1}{x}\right]_{1}^{4} \)
\( \displaystyle =-\frac{1+\log_{e}4}{4}+1 \)
\( \displaystyle =\frac{1}{4}\left(3-\log_{e}4\right) \)