a) ∫arctgx1+x2dx
Substitute arctgx back as y: 12ln(1+x2)+C
Step 1 :Let y=arctgx
Step 2 :Differentiate with respect to x: dydx=11+x2
Step 3 :Substitute y back into the integral: ∫ydy=12y2+C
Step 4 :Substitute arctgx back as y: 12ln(1+x2)+C