22 DIFFERENTIAL AND INTEGRAL CALCULUS. 



(7) -=- (tana#) = a sec 2 ax, fsec'axdx = - tanao?. 



CttJO Cd 



1 



d ( _Jx 



~T l cos (~ 

 dx [ \a 



^ t [ x . _!/#\ -i/ /a: \ 



therefore / -77-= ^r = sin - . or cos 



JV( -O W W 



(differing by JTT). 

 J f ^/icN 



^ S tan = = tan 



^ a*) 



r ^ fo?4\/(^ 2 )) 



-77-= - a . = log ^ - } . 



JV() fo l J 



, 

 therefore 



(There is no absolute necessity for the a in the denomi- 

 nator of the log. It will be found in practice to be simpler 

 to use this form than the one without the a). 



These are the fundamental integrals and should be 

 remembered 5 other integrals can, by various transformations, 

 be made to depend upon some of these ; but every in- 

 tegration is finally an act of the memory, and the different 

 processes of integration only consist in bringing the integral 

 operated upon into a form which the memory recognises. 

 Practice only can render any one expert at the trans- 

 formations required ; but the general formula is as follows : 



Suppose -~ = (f)(x) J i.e. y = J</> (x) dx, then if we can 



simplify $ (x) by putting #=/(z), we shall have -~ -^ -j- ; 



dx w/2 dx 



therefore = = </>{/ (2)} ./' (z) ; and therefore 



