2.6 Improper integrals

Improper integrals


There are two types of improper integrals:
1) Type 1
a) af(x)dx=\int_{a}^{\infty} f(x)dx=lim\limt →\infty atf(x)dx\int_{a}^{t}f(x)dx
b) bf(x)dx=\int_{-\infty}^{b}f(x)dx=lim\limt →-\inftytbf(x)dx\int_{t}^{b}f(x)dx
c) f(x)dx=af(x)dx+af(x)dx\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{a}f(x)dx+\int_{a}^{\infty}f(x)dx

2) Type 2
a) If ff is continuous on [a,b)[a,b) and discontinuous at bb, then:
abf(x)dx=\int_{a}^{b} f(x)dx=lim\limt →b b^-atf(x)dx\int_{a}^{t}f(x)dx
b) If ff is continuous on (a,b](a,b] and discontinuous at aa, then:
abf(x)dx=\int_{a}^{b} f(x)dx=lim\limt →a+ a^+tbf(x)dx\int_{t}^{b}f(x)dx
c) If ff has a discontinuity at cc, where a<c<ba<c<b, then:
abf(x)dx=acf(x)dx+cbf(x)dx\int_{a}^{b} f(x)dx=\int_{a}^{c} f(x)dx+\int_{c}^{b} f(x)dx

If the limits exist and is finite, then it is convergent. Otherwise, it is divergent.
  • 2.
    Type 1 integrals with part a
  • 3.
    Type 1 integrals with part b
  • 5.
    Determining convergence and divergence with type 2 integrals
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Improper integrals

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