DIFFERENTIAL AND INTEGRAL CALCULUS. 39 



3. Assuming that the limiting value of the ratio 



o o 



when n is indefinitely increased, is one of equality, prove 



Wallis' Formula 



TT _ 2 2 4 9 6 a (2rc) a 



2 ~ 2^1 4^1 6 r =T"(2w) s "^i' * 



4. Prove that 



r_ 



J l 



/ 



J 



and generally that 



/"*" c?a; TT 



7 rsTi = -r. -- ^^+i 



J 1 -i- e cosx" 41 1-e**** J 



+ ecosa; 1- 



* dx TT /* dx 



s* 3 " 



- e 



. . 



[1 + COSO?) 



v 





{ 



{transformed by putting tan \x tan \z = / 1 



5. Prove that I $ (x) dx = I <( #) dx, 



J * 



and thence that 



f 71 " n f 71 " r^ 71 " 



^ ^ / 



and that I -. = TT. 



J 1 + sma; 



6. Prove that I** . = .. l ,. cos' 1 (e). 



J 1 + c sin a? v (1 e ) 



and deduce 



I dx = Ja (?r - a), where sin a = e. 



/o sm ^ 



7. Obtain formulas of reduction for 



r a />** r ^ a /v^ 



I Twrz^^. ' "* 



