Improper Integral : A definite integral is an improper integral if either the range of integration is infinite or integrand becomes unbounded at some point.
Proper Integral
A definite integral whose upper and lower limits are finite and the integrand function is finite is finite at any point of the range of integration. That is
\[\int\limits_\alpha ^\beta f (x) dx\]
where \(\alpha ,\beta\) are finite and \(f(x)\) is finite \(\forall x\in \left[\alpha , \beta\right]\)
Evaluation of a proper integral is same as definite integral.
Examples
\(\int\limits_0 ^\pi sin(x) dx\)\(\int\limits_{1} ^{10} (x-1) dx\)
Improper Integrals
If upper or lower or both the limits of an integral are infinite or it’s integrand function tends to infinity at any point in the range of integration then the such an integral is called improper integral.
Therefore improper integrals are divided into following two types
Improper integrals of first type
Examples
\(\int\limits_2^\infty {\frac{1}{{x – 1}}} dx\)\(\int\limits_1^\infty {\frac{1}{{{{x}^2}}}} dx\)
Improper integrals of second type
Examples
\(\int\limits_0^1 {\frac{{\sin (x)}}{x}} dx\)\(\int\limits_\alpha ^\beta {\frac{1}{{{{(\beta – x)}^2}}}} dx\)\(\int\limits_{ – \infty }^0 {{e^x}} dx\)\(\int\limits_{ – \infty }^\infty {\frac{1}{{9 + {x^2}}}} dx\)