Derive the integral for 11+x2
Let say
I=∫11+x2dx------(1)
Now, Put
x=tant⇒t=tan-1xDifferentiatingw.r.txweget⇒1=sec2tdtdx⇒dx=sec2tdt
Putting in equation (1) we get
I=∫11+x2dx=∫sec2t1+tan2tdt⇒I=∫sec2tsec2tdt=∫1dt∵1+tan2t=sec2t⇒I=t+C⇒I=tan-1x+C