The value of definite integral \( \int_{2}^{4}\displaystyle\frac{x}{x^2+1}\,dx \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{1}{2}\log\left(\frac{17}{5}\right) \)
The numerator is half the derivative of the denominator, since \( \frac{d}{dx}(x^2+1)=2x \).
\( \displaystyle \int_{2}^{4}\frac{x}{x^2+1}\,dx \)
\( \displaystyle =\frac{1}{2}\int_{2}^{4}\frac{2x}{x^2+1}\,dx \)
\( \displaystyle =\frac{1}{2}\Big[\log(x^2+1)\Big]_{2}^{4} \)
\( \displaystyle =\frac{1}{2}\left[\log 17-\log 5\right] \)
\( \displaystyle =\frac{1}{2}\log\left(\frac{17}{5}\right) \)