28 



DIFFERENTIAL AND INTEGRAL CALCULUS. 



This integral JV(2aaj x*} dx may also be found directly 

 from the area of the circle. G (fig. 6) the centre, radius 

 OC=a, OM = x, MP=y, then 



y + x- a = a, or y = 



therefore area OMP=f\/(2ax x 2 )dxj corrected so as to 

 vanish when # = 0, = sector OCP&CMP; but whichever 



it be, the whole is vers~M-) + - - <\/(2ax x z ), the 

 2 \dj 2 



second term being positive or negative correctly. 

 Integrals of the form 



dx C dx C dx 



dx f dx [ 



a + b smx ' J a + b cosic ' Ja 



-{-b cosar-f c s'mx 



can be reduced to known forms by the assumption tan \x 

 Thus 



f dx nT? rdz' 

 - = -^ = I = loer^ = log tan \x. 

 Jsniic ] 2z J z 





dx C 2dz (\ - 



[2dz ^_ /!_- 



ji + s 2 : u+x 



J * & 



, M , 1 N 



this may be found from the preceding by putting 

 instead of #, on both sides. 



dx 



( <**. = f 

 ]a + b sina; Ja 



.,/. 



dz 



/ b\v a-tf 

 aU-f-1 +- 

 V a) a 



--?-'/ 



a i> a J , 



tan" 



2 ' 



