[" 183."0^(" sin "^(2x))sin^(-1)sqrt(t)dt+int_(0)^(cos^(2)x)cos^(-1)sqrt(t)" is "(x in[0,pi/2])],[" 1"cet g(x)=int^(x)f(t)" dt,where "f" is such that "(1)/(2)<=f(t)<=1]

int_(0)^(sin^(2)x) sin^(-1) sqrt(t) dt + int_(0)^(cos^(2)x) cos^(-1) sqrt(t) dt=

The value of int_(0)^(sin^(2)x)sin^(-1)sqrt(t)dt+int_(0)^(cos^(2)x)cos^(-1)sqrt(t)dt is

The value of int _0^(sin^2x)sin^(-1)sqrt(t)dt+int _0^(cos^2x)cos^(-1)sqrt(t)dt is

The value of int_(0)^(sin^(2)) sin^(-1)sqrt(t)dt+int_(0)^(cos^(2)x)cos^(-1)sqrt(t)dt , is

f(x)=int_(0)^(x^(2))((sin^(-1)sqrt(t))^(2))/(sqrt(t))dt

(i) If f(x) = int_(0)^(sin^(2)x)sin^(-1)sqrt(t)dt+int_(0)^(cos^(2)x)cos^(-1)sqrt(t) dt, then prove that f'(x) = 0 AA x in R . (ii) Find the value of x for which function f(x) = int_(-1)^(x) t(e^(t)-1)(t-1)(t-2)^(3)(t-3)^(5)dt has a local minimum.

Statement-1: int_(0)^(sin^(2)x) sin^(-1)sqrt(t dt)+int_(0)^(cos^(2)x) cos^(-1)sqrt(t dt)=(pi)/(4) for all x. Statement-2: (d)/(dx) int_(theta(x))overset(psi(x)) f(t)dt=psi'(x)f(psi(x))-psi'(x)f(psi(x))

Statement-1: int_(0)^(sin^(2)x) sin^(-1)sqrt(t )dt+int_(0)^(cos^(2)x) cos^(-1)sqrt(t )dt=(pi)/(4) for all x. Statement-2: (d)/(dx) int_(theta(x))overset(psi(x)) f(t)dt=psi'(x)f(psi(x))-theta'(x)f(theta(x))