Question

Solve each of the following integrals:

${\int }_2^4\frac{x}{x^2+1}dx$ [CBSE 2014]

Answer 1

$$\begin{gathered} {\int }_2^4\frac{x}{x^2+1}dx \\ =\frac{1}{2}{\int }_2^4\frac{2x}{x^2+1}dx \\ =\frac{1}{2}\times {\overline{)\log \left(x^2+1\right)}}_2^4\left[\int \frac{f\text{'}\left(x\right)}{f\left(x\right)}dx=\log f\left(x\right)+C\right] \\ =\frac{1}{2}\left(\log 17-\log 5\right) \\ =\frac{1}{2}\log \left(\frac{17}{5}\right)\left(\log a-\log b=\log \frac{a}{b}\right) \end{gathered}$$