text: # 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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distilabel_metadata.content: # 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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distilabel_metadata.raw_output_q_a_detector_0: The document provides a detailed explanation of the properties of improper integrals, including definitions, formulas, and theorems related to their convergence and divergence. It does not contain any specific questions or answers to questions.
  
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