Answer
\[I=\frac{1}{2}\ln |x^2+1|+C\]
Where $C$ is constant of integration
Work Step by Step
Let \[I=\int\frac{x}{x^2+1}dx\]
\[I=\frac{1}{2}\int\frac{2x}{x^2+1}dx\]
Let $t=x^2+1$ ___(1)
$\Rightarrow dt=2xdx$
\[I=\frac{1}{2}\int\frac{1}{t}dt\]
\[I=\frac{1}{2}\ln |t|+C\]
Where $C$ is constant of integration
From (1)
\[I=\frac{1}{2}\ln |x^2+1|+C\]
Hence,
\[I=\frac{1}{2}\ln |x^2+1|+C\]
