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

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Answer 1 · 2016-04-04T20:08:19

#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#