" i0."quad f(sqrt(x)-(1)/(sqrt(x)))^(2)dx

Evaluate: (i) int(1-x)sqrt(x)dx (ii) int(sqrt(x)+(1)/(sqrt(x)))^(2)dx

Evaluate: (i) int(1-x)sqrt(x)\ dx (ii) int(sqrt(x)+1/(sqrt(x)))^2\ dx

If f(x)=int(2+sqrt(x))/((x+1+sqrt(x))^2)dx and f(0)=0 then the value of If [f(4)] is: (where [.] represents the greatest integer function) .

If int(x(x-1))/((x^(2)+1)(x+1)sqrt(x^(3)+x^(2)+x))dx =(1)/(2)log_(e)|(sqrt(f(x))-1)/(sqrt(f(x))+1)|-tan^(-1)sqrt(f(x))+C, then The value of f(1) is

If int(x(x-1))/((x^(2)+1)(x+1)sqrt(x^(3)+x^(2)+x))dx =(1)/(2)log_(e)|(sqrt(f(x))-1)/(sqrt(f(x))+1)|-tan^(-1)sqrt(f(x))+C, then The value of f(1) is

If int(x(x-1))/((x^(2)+1)(x+1)sqrt(x^(3)+x^(2)+x))dx =(1)/(2)log_(e)|(sqrt(f(x))-1)/(sqrt(f(x))+1)|-tan^(-1)sqrt(f(x))+C, then The value of f(1) is

Evaluate: (i) int((1+sqrt(x))^(2))/(sqrt(x))dx (ii) int sqrt(1+e^(x))e^(x)dx

int_(0)^(a)(sqrt(x))/((sqrt(x)+sqrt(a-x)))dx=(a)/(2)

Let f(x)=(x+1)/(x+2). If int((f(x))/(x^(2)))^((1)/(2))dx=(1)/(sqrt(2))g((sqrt(2f(x))-1)/(sqrt(2f(x))+1))-h((sqrt(f(x))-1)/(sqrt(f(x))+1))

The function y=f(x) is the solution of the differential equation (dy)/(dx)+(x y)/(x^2-1)=(x^4+2x)/(sqrt(1-x^2)) in (-1,1) satisfying f(0)=0. Then int_((sqrt(3))/2)^((sqrt(3))/2)f(x)dx is