Improper Integral An improper integral is an integral with one or more infinite limits and/or discontinuous integrands. Integral is called convergent if the limit exists and has a finite value and divergent if the limit doesn’t exist or has infinite value. This is typically a Calc II topic. Infinite Limit1.()()limtaatfx dxfx dx→∞∞=∫∫2.()()limbbttfx dxfx dx−→−∞∞=∫∫3.()()()ccfx dxfx dxfx dx−−∞∞∞∞=+∫∫∫provided BOTH integrals are convergent. Discontinuous Integrand1.Discont. at a:()()limbbattafx dxfx dx+→=∫∫2.Discont. at b:()()limbtaatbfx dxfx dx−→=∫∫3.Discontinuity at acb<<: ()()()bcbaacfx dxfx dxfx dx=+∫∫∫provided both are convergent. Comparison Test for Improper Integrals : If ()()0fxg x≥≥on [),a∞then, 1.If()afx dx∞∫conv. then()ag x dx∞∫conv. 2.If()ag x dx∞∫divg. then()afx dx∞∫divg. Useful fact : If 0a>then 1apxdx∞∫converges if 1p>and diverges for 1p≤. Approximating Definite IntegralsFor given integral ()bafx dx∫and a n(must be even for Simpson’s Rule) define b anx−∆=and divide ,a binto nsubintervals 01,xx, 12,xx, … , 1,nnxx−with 0xa=and nxb=then, Midpoint Rule : ()()()()***12bnafx dxxfxfxfx≈ ∆+++∫L, *ixis midpoint 1,iixx−Trapezoid Rule : ()()()()()()01212222bnnaxfx dxfxfxfxfxfx−∆≈++ ++++∫LSimpson’s Rule : ()()()()()()()0122142243bnnnaxfx dxfxfxfxfxfxfx−−∆≈++++++∫L
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