24 DIFFERENTIAL AND INTEGRAL CALCULUS. 



The integral I-. - r 77- - 2X is reduced to one 



J(x + a) ^(p + qx + ra?) 



of the preceding by putting x + a = - . 



The forms .. ^ . r . . and ^-y ^ . 2 . are also 

 y(a + oai-f ex ) a + ox-}- ex 



to be integrated by " completing the square," when they 



i 0.1 f p'z + q p'z + q z z 



take the forms -=f, -*-=r- . ^ -^ ; and - - 



' ' ' 



become known forms by the substituting u. It is not 

 at all worth while to complete the integration of all the 

 special cases ; the method by which they are reduced to 

 known forms is what should be attended to. I will take 

 an example of each form. 



(x + 2) + 2 



rt 



dx ? r dx 



-JvK*+2) 2 -lI H 2 Jv{(* + 2r-l} 

 and 



'_ , 



~ Vt( 





therefore 



r^(a; + 4)^_ /tg + 2)< 

 J x' 2 l4^ + 3 ~J ^ 2 -l 



= los (a: + 1} ~ * log(;r 



