# Properties of Improper Integrals

Integrands: $$f$$, $$g$$
Argument (independent variable): $$x$$
Limits of integration: $$a$$, $$b$$, $$c$$, $$n$$
Small real numbers: $$\tau$$, $$\varepsilon$$
1. 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.
2. 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
$${\large\int\limits_a^\infty\normalsize} {f\left( x \right)dx} =$$ $$\lim\limits_{n \to \infty } {\large\int\limits_a^n\normalsize} {f\left( x \right)dx}$$
3. 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.
4. Improper integral with an infinite lower and upper limit
$${\large\int\limits_{ – \infty }^\infty\normalsize} {f\left( x \right)dx} =$$ $${\large\int\limits_{ – \infty }^c\normalsize} {f\left( x \right)dx}$$ $$+\; {\large\int\limits_c^\infty\normalsize} {f\left( x \right)dx}$$
5. 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.
6. 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.
7. 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.
8. 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:
$${\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =$$ $$\lim\limits_{\tau \to 0 + } {\large\int\limits_a^{b – \tau }\normalsize} {f\left( x \right)dx}$$
9. 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
$${\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =$$ $$\lim\limits_{\tau \to 0 + } {\large\int\limits_a^{c – \tau }\normalsize} {f\left( x \right)dx}$$ $$+\;\lim\limits_{\varepsilon \to 0 + } {\large\int\limits_{c + \varepsilon}^b\normalsize} {f\left( x \right)dx}$$
The text passage provides definitions and properties of improper integrals, including formulas and conditions for convergence and divergence. It does not contain any specific mathematical questions or problems to solve.
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# Properties of Improper Integrals

Integrands: $$f$$, $$g$$
Argument (independent variable): $$x$$
Limits of integration: $$a$$, $$b$$, $$c$$, $$n$$
Small real numbers: $$\tau$$, $$\varepsilon$$
1. 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.
2. 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
$${\large\int\limits_a^\infty\normalsize} {f\left( x \right)dx} =$$ $$\lim\limits_{n \to \infty } {\large\int\limits_a^n\normalsize} {f\left( x \right)dx}$$
3. 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.
4. Improper integral with an infinite lower and upper limit
$${\large\int\limits_{ – \infty }^\infty\normalsize} {f\left( x \right)dx} =$$ $${\large\int\limits_{ – \infty }^c\normalsize} {f\left( x \right)dx}$$ $$+\; {\large\int\limits_c^\infty\normalsize} {f\left( x \right)dx}$$
5. 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.
6. 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.
7. 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.
8. 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:
$${\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =$$ $$\lim\limits_{\tau \to 0 + } {\large\int\limits_a^{b – \tau }\normalsize} {f\left( x \right)dx}$$
9. 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
$${\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =$$ $$\lim\limits_{\tau \to 0 + } {\large\int\limits_a^{c – \tau }\normalsize} {f\left( x \right)dx}$$ $$+\;\lim\limits_{\varepsilon \to 0 + } {\large\int\limits_{c + \varepsilon}^b\normalsize} {f\left( x \right)dx}$$
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The text passage provides definitions and properties of improper integrals, including formulas and conditions for convergence and divergence. It does not contain any specific mathematical questions or problems to solve.

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