INTEGRATION OF ALGEBRAIC FRACTIONS 117 



Also 



Then 



- 2 and dx - -dy 



- 8x + 2) 



_5 7 5 



--log e (2*- -____^_^- 2 



f a; n+1 



Knowing the standard form \x n dx = - -, 



J n+ 1 



tion to evaluate \y~ 2 dy and \y~ 3 dy. 



we are in a posi- 



Cfiwe ///. When the denominator of the fraction contains 

 unlike linear factors, some of which are raised to powers. 



x z + 4 



To integrate 



Now 



4 



(x 2 - 4>)(x + 2) 

 x 2 + 4 A 



B 



(a; 2 - 4)(a;+ 2) ( + 2) 2 (aj - 2) as + 2 (^ + 2) a a; - 2 

 and A(a? + 2)(ar - 2) + B(a? - 2) + C(a? + 2) 2 = x 2 + 4 

 When a? = - 2 -4B =8 B=-2 



When x = 2 



When x = - 4A - 2B + 4C 



(# 2 + 4) dx If da: 



16C= 8 

 = 4 



Then 



- 



-2 



:*-2) 



The first and third integrals are such that the numerator is the 

 ifferential coefficient of the denominator, while the second, on 



