DIFFERENTIAL AND INTEGRAL* CALCULUS. 29 



C dx f 2di 



\ 2. / T~-\ i 2\ T 



Ja + ocosx Ja(L + z) + 



dx 2dz _ C dz 



" 



tan'Ms 



each of these will be of another form if b > a. The results 

 given are, however, much the more frequently required, 

 not that these should be remembered, but the method; 

 viz. when sin a? or cos a? occur in the denominator in the 

 first power, put tan \x = z. 

 Should b > a the results are 



f dx 1 (a tan \x + b . 



Ja + b sinx ~ V(^ 2 - a 2 ) ^ i tan Jx + J -f , 

 f dx 1 1 (V(^ + a ) + V(^ ~ a ) tanja? 



J. 



a + o cosx 



The remaining integral I should be 



Ja + b sma? + c cosx 



changed by putting 



b = m cos a, c = m sin a, 



therefore m = V(6 2 + c 2 ), tana =-|L 



and it becomes 



c&c 



f dx or f 



J a -f w sin (a? + a) ] 



sin (a? + a) ] a + m sin (a; + a) " 

 The three integrals 



C dx C dx I dx 



Ja+b sin 2 a? ' Ja + 6 cos s aj ^ Ja + b cos 2 x + c sin* a* 



f <fip 



are essentially of the same form f , and 



J m cos a? + /i sin x 



may be reduced by taking tana? = ^, so that the last in- 

 tegral becomes 



f d* l -if // % \ ^ 



2 = -77 r tan N . / f - } z } , 

 /m + ff*r V(iw) (y \t/ j ' 



