Improper Integrals

Improper Integrals

Identifying Clues

• A definite integral with infinite bound(s)
• A definite integral where the function is discontinuous somehwere on the interval
• The question explicitly states “Find if the integral diverges/what it converges to...”.

Type 1: Infinite Integrals

• ${\displaystyle\int^\infty_a} f(x)\de{x}= \lim\limits_{t\rightarrow \infty}{\displaystyle\int^t_a}f(x)\de{x}$
• ${\displaystyle\int^a_{-\infty}} f(x)\de{x}= \lim\limits_{t\rightarrow -\infty}{\displaystyle\int^a_t}f(x)\de{x}$
• ${\displaystyle\int^\infty_{-\infty}} f(x)\de{x}= \lim\limits_{a\rightarrow -\infty}{\displaystyle\int^0_a}f(x)\de{x} + \lim\limits_{b\rightarrow \infty}{\displaystyle\int^b_0}f(x)\de{x}$

Type 2: Discontinuity at a point

• ${\displaystyle\int^b_a} f(x)\de{x}= \lim\limits_{t\rightarrow b^-}{\displaystyle\int^t_a}f(x)\de{x}$
• ${\displaystyle\int^c_b}f(x)\de{x}=\lim\limits_{t\rightarrow b^+}{\displaystyle\int^c_t}f(x)\de{x}$
• ${\displaystyle\int^c_a}f(x)\de{x}=\lim\limits_{t\rightarrow b^-}{\displaystyle\int^t_a}f(x)\de{x}+\lim\limits_{t\rightarrow b^+}{\displaystyle\int^c_t }f(x)\de{x}$

Convergence