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

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

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

Show that int_(0)^(1)log((1-x)/(x))dx=0

Show that int_(0)^(1)(e^(x))/(1+e^(2x))dx=tan^(-1)(e)-pi/(4)

Evaluate: int_a^b cos x dx

Let f be a continuous function on [a ,b]dot Prove that there exists a number x in [a , b] such that int_a^xf(t)dt=int_x^bf(t)dtdot