Integrands: \(f\), \(g\)

Argument (independent variable): \(x\)

Argument (independent variable): \(x\)

Limits of integration: \(a\), \(b\), \(c\), \(n\)

Small real numbers: \(\tau\), \(\varepsilon\)

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

\({\large\int\limits_{ – \infty }^\infty\normalsize} {f\left( x \right)dx} =\) \( {\large\int\limits_{ – \infty }^c\normalsize} {f\left( x \right)dx} \) \(+\; {\large\int\limits_c^\infty\normalsize} {f\left( x \right)dx} \)
If for some real number \(c\), both of the integrals in the right side are convergent, then the improper integral \(\int\limits_{ – \infty }^\infty {f\left( x \right)dx}\) is also convergent. Otherwise it is 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

\({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \( \lim\limits_{\tau \to 0 + } {\large\int\limits_a^{c – \tau }\normalsize} {f\left( x \right)dx} \) \(+\;\lim\limits_{\varepsilon \to 0 + } {\large\int\limits_{c + \varepsilon}^b\normalsize} {f\left( x \right)dx}\)