Improper Integrals

• ${\int }_a^{b}f(x)dx$ is proper if…
  • Defined and bounded by $[a,b]$
  • $a$ and $b$ are finite
  • The function does not have any discontinuities over $[a,b]$
• Improper integrals
  • If the conditions of a proper integral are not met
  • Two types
    • Either one of the bounds is $\pm \infty$
    • The region defined by $f(x)$ has an infinite discontinuity in $[a,b]$
  • Converges if the limit exists, diverges if not
• Solving when $a$ or $b$ is $\pm \infty$
  • $f$ is cont on $[a,\infty )$ → ${\int }_a^{\infty }f(x)dx={\lim }_{b\to \infty }{\int }_a^{b}f(x)dx$
  • $f$ is cont on $(-\infty ,b]$ → ${\int }_{-\infty }^{b}f(x)dx={\lim }_{a\to -\infty }{\int }_a^{b}f(x)dx$
  • $f$ is cont on $(-\infty ,\infty )$ → ${\int }_{-\infty }^{\infty }f(x)dx={\lim }_{a\to -\infty }{\int }_a^{c}f(x)dx+{\lim }_{b\to \infty }{\int }_c^{b}f(x)dx$
• Solving when region has an infinite discontinuity
  • $f$ is cont on $[a,b)$ and has infinite discount at $b$ → ${\int }_a^{b}f(x)dx=li{m}_{c\to b^{-}}{\int }_a^{c}f(x)dx$
  • $f$ is cont on $(a,b]$ and has infinite discount at $a$ → ${\int }_a^{b}f(x)dx=li{m}_{c\to a^{+}}{\int }_a^{c}f(x)dx$
  • $f$ is cont on $[a,b]$ and has infinite discount at $c$ in $(a,b)$ → ${\int }_a^{b}f(x)dx=li{m}_{d\to c^{-}}{\int }_a^{d}f(x)dx+li{m}_{d\to c^{+}}{\int }_d^{b}f(x)dx$