DIFFERENTIAL AND INTEGRAL CALCULUS. 23 



y=f(f> {f(z}} /' (2) ^ ; i.e. we must substitute for x through- 

 out the whole of the expression under the sign /, putting 

 f (z) dz for dx as well as / (z) for x. As a good example 



f dx 



I g ' _ x , putting s* = , and therefore # = log2, 



take 



<c 



-y- = - ; therefore 

 dz z ' 



dz 



/dx [ z 

 E ^ = J- r = 



This process may also be written, without using a new 

 symbol, as follows : 



The expressions r, , =-. , ^ - 9 . can be 



V(a + bx + car] ' a + bx + cx*' 



reduced to known forms by " completing the square," as 

 if arranging the quadratic involved for solution ; viz. 



_ ( bx tf\ _J^_ 

 \ c 4c'V 4c 



Hence, putting x -f = a, 



" V/- 



cz 



which is of the forms (8) or (10) according as c is positive 

 or negative, and 



/dx f dx 



z + bx + ex* ' J4ac - b 2 " ? 



c 



which is of the forms (4) or (9) according as c is positive 

 or negative. 



