1888.] A. Mukhopadhyay — On Toisson's hitegral. 101 



As the value of the right hand side is well-known, calling J the value of 

 the definite integral in question, it assumes the compact form 



^ (2«)! 



'=i^- j:^^ ^'^ 



This result holds so long as a Z. 1 ; when a y 1, we at once infer from 

 (3) that the result in (4) is to be divided by c?^. 



I now propose to obtain a formula of transformation for the more 

 general integral (2) ; this method has also the advantage of shewing how 

 the indefinite integral itself may be evaluated. Some other integrals 

 which I have arrived at, and which are numbered (6), (7), (9), (10), 

 I have never met with before ; they are, I believe, new. 



§. 2. Transformations of tlie Integral. 

 Consider the general indefinite integral 



_ / sin^^ X dx 



"J (l-2acosx-^a^)'^ 



By putting 



this reduces to 



Now 



P = l + a2, 

 Q = - 2a, 



_ I sin^ X dx 

 J (P+QcosA;r" 



sin^ X dx 



(P + Qcoso;)^ 



2^ ( sin ^ ) ( cos ■^j dx f where 

 / ^^ \^ , ]A = P + Q = (l-a)2 

 (Acos-^l + Bsin^f) (B = P-Q = (l + a)'. 



(x\'^ / oc\^~^'^ 

 sin-1 (cos-j dx 



(x\'^ 

 A + Btan2-j 



/ x\^ / x\^~^ 



2^(tan-j f l + tan2 ^j dx 



(x\^ 

 A + B tan2 -J 



Substituting _ 



