*I\r 



/ 



THE INTEGRALS Jy X ,_ A , AND 



/* 



JVX 2 -A' 



dX 167 



or 



2X 

 A 



X 



A 



VX 2 - A 2 



X+ VX 2 - A 2 



A 



and 



e = log. 



VX 2 -A 2 



Thus 



dx 



Vx 2 + IQx + 36 



, 



ge 



(# + 8) + Vx 2 + I6x + 36 



The 



integral I- 



where X is a linear function of x, will 



IVX 2 - A 2 



give (1) an angle expressed in terms of its hyperbolic cosine or 

 (2) a logarithmic function, and the results can be used as standard 

 forms. 



X 

 A 



or 



(1) 



X + VX 2 - A 2 



(2) 



(b) When the fraction has for a numerator a linear function 

 of x, before proceeding to integration the fraction must be split 

 up into two fractions, the first of which must have for its 

 numerator the differential coefficient of the quadratic expression 

 under the square root. 



8x- 7 

 (b) To integrate 



Now 



Vac 2 + 12a; - 10 

 8x - 7 4 6# + 12 



23 



-f 

 (Ox + 



f 



] Vac 2 + i2x - 10 



- 10 3 Vac 2 + I2x - 10 Vac 2 + 12# - 10 



fdy 

 -7= where y =ac 2 + 120 - 10 



- 2\/ac 2 + I2x - 10 



