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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. Improper integral of a continuous function on the interval [a, infinity)
    4. 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.
    5. Improper integral of a continuous function on the interval (-infinity,b]
    6. Improper integral with an infinite lower and upper limit
      \({\large\int\limits_{ – \infty }^\infty\normalsize} {f\left( x \right)dx} =\) \( {\large\int\limits_{ – \infty }^c\normalsize} {f\left( x \right)dx} \) \(+\; {\large\int\limits_c^\infty\normalsize} {f\left( x \right)dx} \)
    7. Improper integral with an infinite lower and upper limit
      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.
    8. 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.
    9. 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.
    10. 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:
      \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \( \lim\limits_{\tau \to 0 + } {\large\int\limits_a^{b – \tau }\normalsize} {f\left( x \right)dx} \)
    11. Improper integral of a discontinuous function (the point of discontinuity is on the boundary of the interval)
    12. 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
      \({\large\int\limits_a^b\normalsize} {f\left( x \right)dx} =\) \( \lim\limits_{\tau \to 0 + } {\large\int\limits_a^{c – \tau }\normalsize} {f\left( x \right)dx} \) \(+\;\lim\limits_{\varepsilon \to 0 + } {\large\int\limits_{c + \varepsilon}^b\normalsize} {f\left( x \right)dx}\)
    13. Improper integral of a discontinuous function (the point of discontinuity is inside the interval)