Properties of Improper Integrals

Integrands: \(f\), \(g\)
Argument (independent variable): \(x\)

Limits of integration: \(a\), \(b\), \(c\), \(n\)
Small real numbers: \(\tau\), \(\varepsilon\)

The definite integral \(\int\limits_a^b {f\left( x \right)dx}\) is called an improper integral if one of two situations occurs:
– The lower limit of integration \(a\) or the upper limit \(b\) (or both the limits) are infinite;
– The function \(f\left( x \right)\) has points of discontinuity in the interval \(\left[ {a,b} \right]\).
Thus, an improper integral is an integral over an unbounded interval or of an unbounded function.
If \(f\left( x \right)\) is a continuous function on the interval \(\left[ {a,\infty} \right),\)
then the improper integral is expressed through the limit in the form
\({\large\int\limits_a^\infty\normalsize} {f\left( x \right)dx} =\) \(\lim\limits_{n \to \infty } {\large\int\limits_a^n\normalsize} {f\left( x \right)dx} \)

If \(f\left( x \right)\) is a continuous function on the interval \(\left( {-\infty,b} \right],\) then the improper integral is determined by the formula
\({\large\int\limits_{-\infty}^b\normalsize} {f\left( x \right)dx} =\) \( \lim\limits_{n \to -\infty } {\large\int\limits_n^b\normalsize} {f\left( x \right)dx} \)
Note: The improper integrals in formulas \(2\),\(3\) are convergent if the upper and lower limits exist and are finite. Otherwise the improper integrals are divergent.

Improper integral with an infinite lower and upper limit
\({\large\int\limits_{ – \infty }^\infty\normalsize} {f\left( x \right)dx} =\) \( {\large\int\limits_{ – \infty }^c\normalsize} {f\left( x \right)dx} \) \(+\; {\large\int\limits_c^\infty\normalsize} {f\left( x \right)dx} \)

convergent

divergent

Comparison theorems
Let \(f\left( x \right)\) and \(g\left( x \right)\) be continuous functions on the interval \(\left[ {a,\infty} \right)\). Suppose that \(0 \le g\left( x \right) \le f\left( x \right)\) for all \(x\) in \(\left[ {a,\infty} \right)\). Then the following comparison theorems are valid for the improper integrals of the functions \(f\left( x \right)\) and \(g\left( x \right)\):
– If \(\int\limits_a^\infty {f\left( x \right)dx}\) is convergent, then \(\int\limits_a^\infty {g\left( x \right)dx}\) is also convergent.
– If \(\int\limits_a^\infty {g\left( x \right)dx}\) is divergent, then \(\int\limits_a^\infty {f\left( x \right)dx}\) is also divergent.
Absolute convergence
If \(\int\limits_a^\infty {\left| {f\left( x \right)} \right|dx} \) is convergent, then the improper integral \(\int\limits_a^\infty {f\left( x \right)dx} \) is absolutely convergent.
Improper integral of a discontinuous function (the point of discontinuity is on the boundary of the interval)
Let a function \(f\left( x \right)\) be continuous on the interval \(\left[ {a,b} \right)\) but is discontinuous at \(x = b\). Then the following formula is valid:
\({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \( \lim\limits_{\tau \to 0 + } {\large\int\limits_a^{b – \tau }\normalsize} {f\left( x \right)dx} \)

Improper integral of a discontinuous function (the point of discontinuity is inside the interval)
Let a function \(f\left( x \right)\) be continuous for all real numbers \(x\) in the closed interval \(\left[ {a,b} \right]\) except for some point \(c \in \left( {a,b} \right)\). Then
\({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \( \lim\limits_{\tau \to 0 + } {\large\int\limits_a^{c – \tau }\normalsize} {f\left( x \right)dx} \) \(+\;\lim\limits_{\varepsilon \to 0 + } {\large\int\limits_{c + \varepsilon}^b\normalsize} {f\left( x \right)dx}\)

Formulas