Show that `int_a^b(|x|)/x dx=|b|-|a|dot`

int_(a)^(b)e^(x)dx

int_(a)^(b)x^(2)dx

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

If m is the least value (global minimum) and M is the greatest value (global maximum) of the function f(x) on the interval [a , b] (estimation of an integral), then m(b-a)lt=int_a^bf(x)dxlt=M(b-a)dot

Show that : int_0^1(logx)/((1+x))dx=-int_0^1(log(1+x))/x dx

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

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

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

If f(x)is integrable function in the interval [-a,a] then show that int_(-a)^(a)f(x)dx=int_(0)^(a)[f(x)+f(-x)]dx.