162 PRACTICAL MATHEMATICS 



then VA 2 + X 2 = A A/1 + sinh 2 = A cosh 6 



JV 



and X = A sinh 6, 5L- = A cosh 0, or dX = A cosh 

 /DC rA cosh dQ 



JVX 2 + A 2 J A cosh 



_ r 

 J 



= 



T, 1 X 



= sinh- 1 -T- 

 A 



_^^ +6 



Va? + 12* + 48 2\/3 



X 



We also know that sinh = -r- 



A 



nr - ft* p-6\ - 



\JL ~ 1C C "I -T- 



ofl 2.X. 



eJW _ 



. - - 



A 



g _ 



A A 2 A 2 A 2 



^ X VX 2 + A 2 

 A~ A 



X+ A/X 2 + A 2 



and 



Thus 



9^1o g< { X+V f +A2 } 



'(x + 6) + Vx 2 + I2x + 48 



j" dX 



The integral I /y 2 A! wnere X is a linear function of x, will 



give (1) an angle expressed in terms of its hyperbolic sine, or (2) a 

 logarithmic function, and the results can be used as standard 

 forms. 



(D 



VX 2 +A 2 " A 



I 



f rfX f^T J- /-\7-2 A 21 



JVX 2 +A 2 = log H A / ' ' ' (2) 



