How to recognize an improper integral? I think $\displaystyle \int ^{4} _0 \frac{\sin x}{x} dx$ is not an improper integral. Is this true? If not  true, then  how can one recongnise improper integrals in general?

It's not an improper integral because $\dfrac{\sin x} {x}$ has a removable discontinuity at $0$. If $f$ has an infinite essential discontinuity at some point in an interval $I$, then you have an improper integral $\displaystyle\int_If(x) dx$. You also get an improper integral if the interval $I$ where you are integrating is unbounded.
