THE INTEGRALS 



= AND 



169 



(c) When the square root of the quadratic expression appears 

 in the numerator. 



(c) To integrate Vx* - lOx + 20 

 I \/x* - Wx + 20 dr - I V(x -5) 2 -5 dfl 



- f VX 2 -A 2 dX where X =a? -5 and A 2 =5 



put X 2 - A 2 cosh 2 



Then VX 2 - A 2 - Ax/cosh 2 6-1 = A sinh 



TV" 



also X = A cosh 6, -JK = A sinh 0, or dX = A sinh d0 



ttO 



f \/X 2 - A 2 dX = A 2 fsinh 2 dti 



J J 



A 2 f 



= (cosh 20 - 1) d0, since cosh 20 = 2 sinh 2 + 1 

 -{1 sinh 26 -6} 



Then 



{sinh 9 cosh 8 -6} 



I A 2 



cosh- 1 





A 2 /-X-y/X 2 - A 2 (X+VX 2 - 



- --- log - - 



!} 



*} 



\Vx z - 



5f(x-5)Vx z -Wx + 20 , . x-5} 



-{- cosh- 1 7=4 . . . 



2 1 5 \/5 J 



(1) 

 l ' 



2(< 



2\ 



-log e (- 



Y\ 

 The integral 1 A/X 2 A 2 dX can therefore be solved by means 



of two standard forms. 



TX2rfX=4^ 



(1) 

 (2) 



2 V A 2 



cosh- 1 -rl 

 AJ 



-A 2 



