int(1/(25+x^2))dx
dx/d(theta)=5tantheta
dx= 5sec^2theta *(d)theta
int(1/(25+25tan^2theta))* 5sec^2theta*(d)theta
int(1/(25(1+tan^2theta)))* 5sec^2theta*(d)theta
1+tantheta=sec^2theta
int(1/(25(sec^2theta)))* 5sec^2theta*(d)theta
int(1/5)*(d)theta+C
x=5tantheta
x/5 = tantheta
tan^-1(x/5)=theta
Plug in:
=1/5 tan^-1 (x/5) +C