Suppose we define the definite integral using the following formula `int_a^b f(x)dx=(b-c)/2(f(a)+f(b))` , for more accurate result for `c in (a, b)F(c)=(b-c)/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)+f(2f(c))` then `int_0 ^(pi/2) sinxdx`

Suppose we define the definite integral using the following formula int_a^b f(x)dx=(b-a)/2 (f(a)+f(b)) , for more accurate result for c in (a,b), F(c)=(c-a)/2 (f(a)+f(c))+(b-c)/2(f(b)+f(c)) and when c=(a+b)/2, int_a^b f(x)dx=(b-a)/4(f(a)+f(b)+2f(c)) In the above comprehension, if f(x) is a polynomial and lim_(trarra) (int_a^t f(x)dx-(t-a)/2 [f(t)+f(a)])/(t-a)^3=0 for all a then the degree of f(x) can at most be (A) 1 (B) 2 (C) 3 (D) 4

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

Let the definite integral be defined by the formula int_(a)^(b)f(x)dx=(b-a)/2(f(a)+f(b)) . For more accurate result, for c epsilon (a,b), we can use int_(a)^(b)f(x)dx=int_(a)^(c)f(x)dx+int_(c)^(b)f(x)dx=F(c) so that for c=(a+b)/2 we get int_(a)^(b)f(x)dx=(b-a)/4(f(a)+f(b)+2f(c)) . If f''(x)lt0 AA x epsilon (a,b) and c is a point such that altcltb , and (c,f(c)) is the point lying on the curve for which F(c) is maximum then f'(c) is equal to

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

Evaluate each of the following integral: int_a^b(f(x))/(f(x)+f(a+b-x))dx

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

If f(a+b-x)=f(x), then prove that int_(a)^(b)xf(x)dx=(a+b)/(2)int_(a)^(b)f(x)dx