38 DIFFERENTIAL AND INTEGRAL CALCULUS. 



-IA -n A.I. [** 

 10. Prove that 



J Q 1 + ecoso; 



and 



4* dx -e . 1 







^ 



( To integrate -. ^ first differentiate ) 



\ (1+6 cosx) 1 + e cosov 



QUESTIONS ON INTEGRAL CALCULUS. II. 



1. Find the value of 



r w 



j i 



generally of I ^{ ab + (a + b) x x*} dx, 



J a 



and of I x V{- 



Ja 



jPut a 



the answers are TT, JTT (b a)' 2 , (b + a) (b a) 2 [- . 

 Prove also that 



| x(x- 1) (x-2)(x-3)...(x-2n) da 



2. Prove the following formula of reduction : 



C a n-l r a 



(1) I x n V(* - *) dx = ^ ; g a a I x""* V( a - 



* CL Q 



r n j( *}d 2?t+i r "- 1 



J w + 2 J 



(3) / sin n ic^c = ^ - ( *Bu?-*xdx 



Jo n J o 



, [ V ' n 7 J ' 



= | I sm icoa; = / cos xdx 



J A 



