Improper Integrals

Abstract

The need to extend the definition of the integral beyond the limits imposed in Chapter 4 becomes apparent quite quickly, sometimes even without noticing if we are not too careful. For instance, if we want to integrate f (x) = x^p over [0,1] then we can write uncritically

$$ \int_0^1 {x^p dx = \left[ {\frac{{x^{p + 1} }} {{p + 1}}} \right]_0^1 = \frac{1} {{p + 1}}\left( {1^{p + 1} - 0^{p + 1} } \right) = \frac{1} {{p + 1}}.} $$