Prove that for `0 < x < pi/2`, sinx < x < tan x. Deduce that `underset(xrarr0)Lt(sinx)/x=1`.

Prove that for x in [0, (pi)/(2)], sin x + 2x ge (3x(x + 1))/(pi) .

Prove that for any positive integer k , (sin2k x)/(sinx)=2[cosx+cos3x++cos(2k-1)x]dot Hence, prove that int_0^(pi/2)sin2x kcotxdx=pi/2dot

Prove that f(x) = [x], 0 lt x lt 3 is not differentiable at x = 1 but x = 2 .

Let f (x) =(ax+b)/(x+1), lim_(x rarr 0)f(x)=2 and lim_(x to oo)f(x)=1 , Prove that f (- 2) = 0.

Prove that : int_0^pi (xtanx)/(secxcosecx)dx =pi^2/4 .

Prove that : ""∫_0 ^(pi/2)sin2xlogtanxdx=0

Prove that following: underset(0)overset(1)int|x-5|dx

Prove that following: underset(0)overset(4)int|x-1| dx

Prove that following: underset(0)overset(pi)int|cosx|dx

Prove that : C_(0)-3C_(1)+5C_(2)- ………..(-1)^n(2n+1)C_(n)=0