Calculus Small Changes, Big Ideas

Section 9.3 Improper integrals

• Infinite intervals:

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  \begin{align*} \int_a^\infty f(x)\dd{x} \amp = \limit{b\to\infty}\int_{a}^{b} f(x)\dd{x} \\ \int_{-\infty}^{b} f(x)\dd{x} \amp = \limit{a\to-\infty}\int_{a}^{b} f(x)\dd{x} \\ \int_{-\infty}^{\infty} f(x)\dd{x} \amp = \int_{-\infty}^{a} f(x)\dd{x} + \int_{a}^{\infty} f(x)\dd{x} \end{align*}
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• Discontinuous integrands:

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  If \(f\) is continuous on \([a,b)\) and discontinuous at \(b\text{,}\) then

  \begin{equation*} \int_{a}^{b} f(x)\dd{x}=\limit{t\to b^-}\int_{a}^{t} f(x)\dd{x}\text{.} \end{equation*}

  If \(f\) is continuous on \((a,b]\) and discontinuous at \(a\text{,}\) then

  \begin{equation*} \int_{a}^{b} f(x)\dd{x}=\limit{t\to a^+}\int_{t}^{b} f(x)\dd{x}\text{.} \end{equation*}

  If \(f\) has a discontinuity at \(c\text{,}\) where \(a \lt c \lt b\text{,}\) then

  \begin{equation*} \int_{a}^{b} f(x)\dd{x}=\int_{a}^{c} f(x)\dd{x}+\int_{c}^{b} f(x)\dd{x} \end{equation*}

  as long as the two indefinite integrals converge.

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• The \(p\)-integral \(\DS\int_{1}^{\infty}\dfrac{1}{x^{p}}\dd{x}\) converges if and only if \(p>1\text{.}\)

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• The \(p\)-integral \(\DS\int_{0}^{1}\dfrac{1}{x^{p}}\dd{x}\) converges if and only if \(p<1\text{.}\)

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• Gamma function:

  \begin{equation*} \Gamma(x)=\int_{0}^{\infty} t^{x-1}\ee^{-t}\dd{t} \end{equation*}

  For any integer \(n\) we have \(\Gamma(n)=(n-1)!\text{.}\)

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• Some notational conventions I’d like to introduce (assuming the limits exist):

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  \begin{align*} f(\infty) \amp = \limit{x\to\infty} f(x) \\ f(-\infty) \amp = \limit{x\to-\infty} f(x) \\ f(a^{+}) \amp = \limit{x\to a^+} f(x) \\ f(a^{-}) \amp = \limit{x\to a^-} f(x) \end{align*}
  These could make writing out the improper integrals somewhat cleaner.

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