Given that `f` satisfies `|f(u)-f(v)|lt=|u-v|` for u and v in `[a , b]dot` Then `|int_a^bf(x)dx-(b-a)f(a)|lt=` (a)`((b-a))/2` (b) `((b-a)^2)/2` (c)`(b-a)^2` (d) none of these

int_(a)^(b) f(x) dx =

Prove that int_(a)^(b)f(x)dx=(b-a)int_(0)^(1)f((b-a)x+a)dx

Let f(x)=|x-1|dot Then (a) f(x^2)=(f(x))^2 (b) f(x+y)=f(x)+f(y) (c) f(|x|)-|f(x)| (d) none of these

If f(x) is monotonic differentiable function on [a , b] , then int_a^bf(x)dx+int_(f(a))^(f(b))f^(-1)(x)dx= (a) bf(a)-af(b) (b) bf(b)-af(a) (c) f(a)+f(b) (d) cannot be found

f(x) is a continuous function for all real values of x and satisfies int_n^(n+1)f(x)dx=(n^2)/2AAn in Idot Then int_(-3)^5f(|x|)dx is equal to (19)/2 (b) (35)/2 (c) (17)/2 (d) none of these

If alpha,beta(beta>alpha), are the roots of g(x)=a x^2+b x+c=0 and f(x) is an even function, then int_alpha^betae^(f((g(x))/(x-alpha)))/(e^(f((g(x))/(x-alpha)))+e^(f((g(x))/(x-beta))))= (a) |b/(2a)| (b) (sqrt(b^2-4a c))/(|2a|) (c) |b/a| (d) none of these

Given a real-valued function f which is monotonic and differentiable. Then int_(f(a))^(f(b))2x(b-f^(-1)(x))dx=

C F is the internal bisector of angle C of A B C , then C F is equal to (2a b)/(a+b)cosC/2 (b) (a+b)/(2a b)cosC/2 (c) (b sinA)/(sin(B+C/2)) (d) none of these

If f(x)=|[a,-1, 0],[a x, a,-1],[a x^2,a x, a]|,t h e nf(2x)-f(x) is divisible by (a) a (b) b (c) c ,d ,e (d) none of these

If f^(prime)(x)=f(x)+int_0^1f(x)dx ,gi v e nf(0)=1, then the value of f((log)_2 2) is (a) 1/(3+e) (b) (5-e)/(3-e) (c) (2+e)/(e-2) (d) none of these