#int dx/(x^2+1)^2 #
Use #x=tan(a)#
#dx=sec^2(a)da#
#intdx/(x^2+1)^2=int (sec^2(a)da)/(1+tan^2a)^2#
Use the identity # 1+tan^2(a)=sec^2(a)#
#intdx/(x^2+1)^2=int (sec^2(a)da)/sec^4(a)#
#=int (da)/sec^2(a)#
#=int cos^2(a) da#
#=int ((1+cos(2a))/2 )da#
#=(1/2)(int (da)+int cos(2a)da)#
#=(1/2)(a+sin(2a)/2)#
#=(1/2)(a+(2sin(a)cos(a))/2)#
#=(1/2) (a+sin(a).cos(a))#
we know that #a=tan^-1(x)#
#sin(a)=x/(sqrt(1+x^2)#
#cos(a)=x/(sqrt(1+x^2#
#int dx/(x^2+1)^2=
(1/2) (tan^-1(x) +sin(sin^-1(x/(sqrt(1+x^2)))cos(cos^-1(1/(sqrt(1+x^2))))#
#=(1/2) (tan^-1(x)+(x/(sqrt(1+x^2))1/sqrt(1+x^2))#
#=(1/2) (tan^-1(x)+x/(1+x^2))#