`int_a^bf(x)dx=`

Definite integral of any discontinuous or non differentiable function is normally done by the usage of the property int_a^bf(x)dx=int_a^bf(x)dx+int_c^bf(x)dx ,where c in(a,b) is the point of discontinuity or non diffrentiability The value of I=int_1^100[sec^(-1)x]dx ,(where[.]denotes greatest integer function)is equal to

Suppose in the definite integral int_a^b f(x) dx the upper limit b->oo, then to obtain the value of int_a^bf(x) dx , we may say that int_a^bf(x)dx=lim_(k->oo)int_a^k dx, where k > a. if f(x)->oo as x ->a or x->b, then the value of definite integral int_a^bf(x)dx is lim_(h->0) int_(a+h)^b f(x) dx. If this limit the value of the limit is defined as the value of integral. This should be noted that f (x)should not have any other discontinuity in [a, b] otherwise this will lead to errorous solution.

STATEMENT 1: If f(x) is continuous on [a , b] , then there exists a point c in (a , b) such that int_a^bf(x)dx=f(c)(b-a) STATEMENT 2: For a < b , if ma n dM are, respectively, the smallest and greatest values of f(x)on[a , b] , then m(b-a)lt=int_a^bf(x)dxlt=(b-a)Mdot

STATEMENT 1: If f(x) is continuous on [a , b] , then there exists a point c in (a , b) such that int_a^bf(x)dx=f(c)(b-a) STATEMENT 2: For a < b , if ma n dM are, respectively, the smallest and greatest values of f(x)on[a , b] , then m(b-a)lt=int_a^bf(x)dxlt=(b-a)Mdot

STATEMENT 1: If f(x) is continuous on [a , b] , then there exists a point c in (a , b) such that int_a^bf(x)dx=f(c)(b-a) STATEMENT 2: For a

Consider a real valued continuous function f(x) defined on the interval [a, b). Which of the followin statements does not hold(s) good? (A) If f(x) ge 0 on [a, b] then int_a^bf(x) dx le int_a^bf^2(X)dx (B) If f (x) is increasing on [a, b], then f^2(x) is increasing on [a, b]. (C) If f (x) is increasing on [a, b], then f(x)ge0 on (a, b). (D) If f(x) attains a minimum at x = c where a lt c lt b, then f'(c)=0.

If | int_a ^b f(x) dx| = int_a ^b |f(x)| dx, a lt b , then f(x) = 0 has

If for nonzero x ,af(x)+bf(1/x)=1/x-5, where a!=b then int_1^2f(x)dx=

Let f(x) is continuous and positive for x in [a , b],g(x) is continuous for x in [a , b]a n dint_a^b|g(x)|dx >|int_a^bg(x)dx| STATEMENT 1 : The value of int_a^bf(x)g(x)dx can be zero. STATEMENT 2 : Equation g(x)=0 has at least one root for x in (a , b)dot

Let f(x) is continuous and positive for x in [a , b],g(x) is continuous for x in [a , b]a n dint_a^b|g(x)|dx >|int_a^bg(x)dx| STATEMENT 1 : The value of int_a^bf(x)g(x)dx can be zero. STATEMENT 2 : Equation g(x)=0 has at least one root for x in (a , b)dot