The value of \(\int_{2}^{4}\displaystyle\frac{x}{x^2+1}\,dx\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{1}{2}\log_e\left(\frac{17}{5}\right)\)
The numerator is half the derivative of the denominator, so substitute
\(u = x^2 + 1,\quad du = 2x\,dx\)
The limits change: \(x = 2\) gives \(u = 5\), and \(x = 4\) gives \(u = 17\). Then
\( \displaystyle \int_{2}^{4}\frac{x}{x^2+1}\,dx = \frac{1}{2}\int_{5}^{17}\frac{du}{u} \)
\( \displaystyle = \frac{1}{2}\Big[\log_e u\Big]_{5}^{17} \)
\( \displaystyle = \frac{1}{2}\left(\log_e 17 - \log_e 5\right) = \frac{1}{2}\log_e\left(\frac{17}{5}\right) \)