Improper Integrals (Sample 6) - Chapter 7.8 Improper...

This preview shows page 1 - 2 out of 6 pages.

Chapter 7.8 Improper Integrals If 𝑓is continuous over [?, ?]and ∫ 𝑓(𝑥)?𝑥 = 𝐹(𝑥) + 𝐶, then ∫ 𝑓(𝑥)?𝑥 = 𝐹(?) − 𝐹(?).??When is an integral improper? ∫ 𝑓(𝑥)?𝑥 = 𝐹(?) − 𝐹(?)??Case 1.Unbounded interval of integration Case 2.𝑓has an infinite discontinuity over the interval of integration. Infinite discontinuity refers to an asymptotic behaviour of a function. Integrals over unbounded intervals Unbounded Intervals (−∞, ?]𝑓(𝑥)?𝑥?−∞[?, +∞)𝑓(𝑥)?𝑥+∞?(−∞, +∞)𝑓(𝑥)?𝑥+∞−∞Assume that,𝑓is continuous over the interval of integration. 𝑓(𝑥)?𝑥+∞?= lim𝑡→∞∫ 𝑓(𝑥)?𝑥𝑡?𝑓(𝑥)?𝑥?−∞=lim𝑡→−∞∫ 𝑓(𝑥)?𝑥?𝑡𝑓(𝑥)?𝑥+∞−∞= ∫𝑓(𝑥)?𝑥0−∞+ ∫𝑓(𝑥)?𝑥+∞0If the respective limit(s) exists and is finite, the improper integral is convergent. If the limit(s) does not exist or is infinite, the improper integral is divergent. Infinite discontinuity over the integral of integration Assume that 𝑓is continuous over (?, ?] and lim𝑥→?+𝑓(𝑥) = ±∞.∫ 𝑓(𝑥)?𝑥??= lim𝑡→?+∫ 𝑓(𝑥)?𝑥?𝑡Assume that 𝑓is continuous over [?, ?) and lim𝑥→?𝑓(𝑥) = ±∞.∫ 𝑓(𝑥)?𝑥?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture