Recap of definite integrals

We will frequently have to evaluate

$$F(x):={\int }_{-\mathrm{\infty }}^{x}f(s)ds.$$

For any particular value of $x$ (e.g. $x=2$) this is the definite integral that you know. It is NOT the same as the indefinite integral, i.e.

$$F(x)\ne \int f(x)dx.$$

For example, suppose

Then for , $F(x)=0$; when $x\ge 1$,

$${\int }_{-\mathrm{\infty }}^{x}f(s)ds={\int }_1^{x}f(s)ds={\left[-\frac{1}{s}\right]}_1^x=1-1/x.$$

Whereas, for $x\ge 1$

$$\int f(x)dx=-1/x+c,$$

which gives a whole family of functions (including the one we want) depending on $c$.

One of the most frequent single mistakes for beginner students in probability is evaluating an indefinite integral with $c=0$ when they should have been evaluating a definite integral.