42 DIFFERENTIAL AND INTEGRAL CALCULUS. 



and the differential coefficient of this with respect to a 

 will be (1) + (2)., V(4a)4-(3), \/() or 7 V() which is 

 obviously true. 



ra 

 \/(ax) dx ^a 2 , the 

 a 



complete differential coefficient with respect to a is 



or 



which is obviously correct. 



The second differential coefficient is often calculated, 

 but it is better to use the above again when necessary. 

 A good example of such integration is the following : 

 we have 



* dx 



r 



J l 



+ esnx 



r 



2 J i- 



-i 



2t! 



~ [iTT-tan- 1 !-^- 



-U-. (-JTT - siif'e) = -jj- -=r cos' 1 (e) ; 



whence integrating both sides with respect to ^, which has 

 no concern with the limits, 



r 



J 



logfl -f e smx) ( -if M2 , i- 



&v . - -dx = - {cos Ve)} 2 -h A, 



sma; 



K being independent of e, and when e = 0, the left hand 

 vanishes, therefore K= (i^) 2 , whence if we put e = sin 0, 

 we have 



r og + snBnx ^ = . , _ ( , ^, = ^ _ ff 

 L sin* 



