What is the integral of ##int ( 1 / (25 + x^2) ) dx ##?

##int(1/(25+x^2))dx=1/5 tan^-1 (x/5) +C##

##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##

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