a) \( \int \frac{\operatorname{arctg} x}{1+x^{2}} d x \)

Step 1 ：Let \( y = \operatorname{arctg} x \)

Step 2 ：Differentiate with respect to x: \(\frac{d y}{d x}=\frac{1}{1+x^2} \)

Step 3 ：Substitute y back into the integral: \(\int y d y = \frac{1}{2}y^2 + C \)

Step 4 ：Substitute \( \operatorname{arctg} x \) back as y: \( \frac{1}{2}\ln(1+x^2) + C \)