Evaluate: `int_0^(sin^2x) sin^-1sqrt(t)dt+int_0^(cos^2x) cos^-1sqrt(t)dt`

Let f(x)=int_0^(sin^2x) sin^-1(sqrt(t))dt+int_0^(cos^2x) cos^-1(sqrt(t))dt , then (A) f(x) is a constant function (B) f(pi/4)=0 (C) f(pi/3)=pi/4 (D) f(pi/4)=pi/4

Let f(x)=int_0^(sin^2x) sin^-1(sqrt(t))dt+int_0^(cos^2x) cos^-1(sqrt(t))dt , then (A) f(x) is a constant function (B) f(pi/4)=0 (C) f(pi/3)=pi/4 (D) f(pi/4)=pi/4

(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))-theta'(x)f(theta(x))

Prove that: y=int_(1/8)^(sin^2x)sin^(-1)sqrt(t)dt+int_(1/8)^(cos^2x)cos^(-1)sqrt(t) , where 0lt=xlt=pi/2 , is the equation of a straight line parallel to the x-axis. Find the equation.

int sqrt(t)dt

int(1)/(sqrt(t))dt

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

Evaluate: int(sin^(-1)sqrt(x)-cos^(-1)sqrt(x))/(sin^(-1)sqrt(x)+cos^(-1)sqrt(x))\ dx

Evaluate: int(sin^(-1)sqrt(x)-cos^(-1)sqrt(x))/(sin^(-1)sqrt(x)+cos^(-1)sqrt(x))dx