Indeterminate integral of `f(x)` (also called antiderivative) is function (more precisely a set of functions) such that `F'(x)=f(x)`. It is written as `F(x)=int f(x)dx` or `F(x)=int_a^x f(t)dt`.
The latter is definite integral with variable upper limit while the former is just different way of writing the same.
Example: `int x^ndx=x^(n+1)/(n+1)+C` where `C in RR`
Improper integral is similar to definite integral `int_a^b f(x)dx` but with one difference. In case of improper integral one or both limits are infinite e.g. `int_a^infty f(x)dx`
Example: ` `
`int_-infty^infty e^-xdx=lim_(a->infty)int_-a^a e^-xdx=lim_(a->infty)-e^-x|_(-a)^a=`
As you can see indefinite integral is set of functions while improper and definite integrals are actually numbers (area under the curve `f(x)`).