1.If a function doesn't "have" indefinite integral , then it doesn't "have" definite integral too because they are both same thing except the constant gets cancelled in definite integral and we get a definite value for 2 given limits . Now some functions (like $\ln(\ln x),x^x,\sin(\sin x)$ ) can not be integrated . These are called non-elementary functions. As far as I know , although they can't be integrated , their integral value between 2 limits are

calculated approximately . I dont know how though(haven't studied yet!

).

Integral's geometric outcome is calculating area covered by $f(x)$ with $X$ axis . Even we can't calculate non-elementary function's integral directly , that doesn't mean the function doesn't cover any area . We calculate this

approximately .

2. While starting derivative chapters you may learn that $\displaystyle \frac{d}{dx}(\ln x)=\frac{1}{x}$ ,

as Integration is the reverse process of differentiation [I think here you will get a nice

explanation about reverse relation] ,

$\displaystyle \int \frac{1}{x}dx=\ln x+c$ .