DIFFERENTIAL AND INTEGRAL CALCULUS. 27 



therefore 



f // - M\ 2 -\ X f / 



|V(a x > ax = x y(a x ) + a sin - |v 

 j J 



and therefore = jo? *J(a 2 - x 2 ) 4 a* sin" 1 f - 



As this is one of the most frequently occurring integrals, 

 it may be well to mention that the result may be obtained 

 directly by considering the area of a circle, centre 0, 

 radius = a, P (fig. 5) any point of the circle, OMx, 

 MP=y, then x z -y 2 = d 2 , or MP=^(a 2 x z )j and the 

 area AOMP = j*J(d 2 x 2 ) dx, corrected so as to vanish 

 when x vanishes. But this area = sector A OP + &POM, 



fx\ 

 = \ a 2 sin" 1 ! - J -f | x \/(a 2 - x 2 ), whence 



and no constant will be wanted if the integral vanish when 

 x vanishes. 



Another important integral f^/(2ax x z ) dx is at once 

 reduced to this by writing it JV{a 2 (x a)' 2 } dx, which is 



therefore =^(x-a) \/{a'- (x-a)' 2 } + - sin' 1 ( - - ) . but in 



& \ a J 



this form the integral will not vanish when x vanishes, 

 but when x = a ; and as it is generally better to take the 

 integral of such a form as to vanish with x, we must add 

 such a constant to this as will make it do so. Now when 



x = 0, the above result becomes -- , and therefore - 



4 ' 4 



should be added, and the result will be 





 sin 



or "V^-o^ + ^vers-Y-V 



2 2 \aj 



(For if iTr + sm- 1 , = sin (d - 



or x = a (1 cos#) = a vers0). 



