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

int_(a)^(b)(f(x))/(f(x)+f(a+b-x))dx=

int_(a)^(b)(f(x))/(f(x)+f(a+b-x))dx=

Given a real valued fuction f(x) which is monotonic and differentiable , prove that for any real number a and b, int_(a)^(b){f^(2)(x)-f^(2)(a)} dx = int_(f(a))^(f(b))2x{b-f^(-1)(x)} dx

Given a real valued fuction f(x) which is monotonic and differentiable , prove that for any real number a and b, int_(a)^(b){f^(2)(x)-f^(2)(a)} dx = int_(f(a))^(f(b))2x{b-f^(-1)(x)} dx

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

For every function f (x) which is twice differentiable , these will be good approximation of int_(a)^(b)f(x)dx=((b-a)/(2)){f(a)+f(b)} , for more acutare results for cin(a,b),F( c) = (c-a)/(2)[f(a)-f( c)]+(b-c)/(2)[f(b)-f( c)] When c= (a+b)/(2) int_(a)^(b)f(x)dx=(b-a)/(4){f(a)+f (b)+2 f ( c) }dx Good approximation of int_(0)^(pi//2)sinx dx , is

For every function f (x) which is twice differentiable , these will be good approximation of int_(a)^(b)f(x)dx=((b-a)/(2)){f(a)+f(b)} , for more acutare results for cin(a,b),F( c) = (c-a)/(2)[f(a)-f( c)]+(b-c)/(2)[f(b)-f( c)] When c= (a+b)/(2) int_(a)^(b)f(x)dx=(b-a)/(4){f(a)+f (b)+2 f ( c) }dx If lim_(t toa) (int_(a)^(t)f(x)dx-((t-a))/(2){f(t)+f(a)})/((t-a)^(3))=0 , then degree of polynomial function f (x) atmost is

For every function f (x) which is twice differentiable , these will be good approximation of int_(a)^(b)f(x)dx=((b-a)/(2)){f(a)+f(b)} , for more acutare results for cin(a,b),F( c) = (c-a)/(2)[f(a)-f( c)]+(b-c)/(2)[f(b)-f( c)] When c= (a+b)/(2) int_(a)^(b)f(x)dx=(b-a)/(4){f(a)+f (b)+2 f ( c) }dx If lim_(t toa) (int_(a)^(t)f(x)dx-((t-a))/(2){f(t)+f(a)})/((t-a)^(3))=0 , then degree of polynomial function f (x) atmost is