\int \frac{x}{1+x^{2}}dx=\frac{1}{2}\int \frac{dt}{1+t}= \frac{1}{2}\int \frac{dt}{1^{2} + (\sqrt{t})^{2}}=\frac{1}{2}arctg\sqrt{t}=\frac{1}{2}arctg\sqrt{x^{2}}= \frac{1}{2}arctgx + C