CHAPTER XI 



92. The standard forms for Isin (ax + b) dx and I cos (ax + b) dx. 



If y = cos (ax + b), then -^- = asm (ax+b) 



Hence la sin (ax + b) dx = cos (ax + b) 



f . cos (ax + b) 

 and Ism (ax+b) dx = * .... (1) 



Also if y = sin (ax + b), then -j^ = a cos (ax + b) 



Hence 1 a cos (ax + b) dx = sin (ax + b) 



C sin (ax + b) 

 and cos (ax + b) dx = * L (2) 



93. The Hyperbolic Functions. 



If y = cosh (oo! + 6), then ^ = a sinh (o# + b) 



Hence la sinh (ax + b) dx = cosh (ax + b) 



f . cosh (ax + b) 

 and sinh (ax+ b) dx = * '- (3) 



Also if y = sinh (ax + b), then -j- = a cosh (ax + b) 



Hence la cosh (ax + b) dx = sinh (ax + b) 



f , ... sinh (ax + b) 

 and Icosh (ax + b) dx = ! (4) 



94. These results may be applied to the integration of algebraic 

 fractions, the denominators of which consist of the square root 

 of a quadratic expression. 



It has been shown in the previous chapter that an expression 

 of the form ax 2 + bx+ c reduces down to one of the three forms 

 a(A 2 - X 2 ), a(X 2 + A 2 ), or a(X 2 - A 2 ), where X is a linear function 

 of x and the consideration of these integrals depends upon the 

 particular form the denominator takes. 



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