`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.

int_(0)^(1)t^(2)sqrt(1-t)*dt

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

Evaluate the following integrals using properties of integration : int_(0)^(sin^(2)x) sin^(-1) sqrt(t) dt + int_(0)^(cos^(2)x)sqrt(t) dt

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))

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

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))