152 



PRACTICAL MATHEMATICS 



90. Case II. When the denominator can be expressed as the 

 sum of two squares that is, it reduces to the form a(X 2 + A 2 ) 



(a) To integrate 



Then x z + 14a? + 60 = x 2 + 14# +49+11 



= (x + 7) 2 + 11 

 f dx f dx 



1 , .X 



= -r- tan- 1 -r- 



A A 



= -7=1 tan- 1 



Vn Vii 



(&) When the numerator of the fraction is of the first degree in x, 

 the fraction should be expressed as the sum or difference of two 

 fractions, the first being formed so that its numerator is the 

 differential coefficient of the denominator. 



7x+3 

 To integrate 



Now 



Also 



x 2 + I2x + 52 



7 2z + 12 



45 



+12* +52 2 # 2 + 120 +52 x 2 + 12^+52 

 dx f dx 



J( 



120+52 J(0+ 6) 2 +16 



where X = x +6 and A 2 = 16 



f 



X 2 + A 2 



= T tan - 1 T 

 A A 



- tan" 1 



4 



