Show that, `int_(0)^(n pi+v)|sin x|dx=2n+1-cosv`, where n is a positive integer and `0 le v le pi`.

Show that int_0^(npi+v)|sinx|dx=2n+1-cosv , where n is a positive integer and , 0<=vltpi

int (dx)/(1+nsqrt(x+1)) , n is a positive integer.

Using the formula of differentiation of product functions show that (d)/(dx)(x^(n))=nx^(n-1) , where n is a positive integer.

Evaluate int_(0)^(npi+t)(|cosx|+|sinx|)dx, where n epsilonN and t epsilon[0,pi//2] .

If U_n=int_0^pi(1-cosnx)/(1-cosx)dx , where n is positive integer or zero, then show that U_(n+2)+U_n=2U_(n+1)dot Hence, deduce that int_0^(pi/2)(sin^2ntheta)/(sin^2theta)=1/2npidot

Show that, int_(0)^(pi)xf(sinx)dx=(pi)/(2)int_(0)^(pi)f(sinx)dx .

Prove that (""^n C_0)/x-(""^n C_1)/(x+1)+(""^n C_2)/(x+2)-.....+(-1)^n(""^n C_n)/(x+n)=(n !)/(x(x+1) . . . (x-n)), where n is any positive integer and x is not a negative integer.

sin theta + sin 5 theta = sin 3theta (0 le theta le 2pi)

Let 0 lt A_(i) lt pi for i = 1,2,"……"n . Use mathematical induction to prove that sin A_(1) + sin A_(2)+ "….." + sin A_(n) le n sin ((A_(1) + A_(2) + "……" + A_(n))/(n)) where n ge 1 is a natural number. [You may use the fact that p sin x + (1-p) sin y le sin [px+(1-p)y] , where 0 le p le 1 and 0 le x , y le pi ]

Given (sin2kx)/(sin x)=2[cos x+cos 3x+…+ cos(2k-1)x] , where k is a positive integer, show that, int_(0)^((pi)/(2))sin2 kx cotx dx=(pi)/(2) .