Improper Integrals – How to Integrate Through an Infinity

What do you do when the function you're integrating shoots off to infinity inside the interval? Simpson's rule needs a value there and can't get one — patch it, and halving h improves the answer only 1.42 times instead of 16. The fix: split off the part that blows up, integrate its Taylor polynomial exactly, and hand the smooth leftover to Simpson's rule. Same 5 points, an error of about 0.0000106. In this lecture: • why Simpson's rule fails at a singularity • splitting f into a Taylor part (finite, exact) and a smooth leftover • the method in four steps • the other kinds of improper integral: infinite in the middle, infinite intervals • worked example: an infinite interval, with t = 1/x • three exercises, worked in full Worked examples and exercises from Burden & Faires, Numerical Analysis (9th ed.), Section 4.9. CHAPTERS 0:00 A function that goes to infinity 3:36 Why Simpson's rule fails here 4:33 Split off the infinite part 6:35 The easy part: power rule 9:02 Simpson on the leftover 14:14 The method in four steps 15:39 Other kinds of improper integral 17:06 Worked example: an infinite interval 23:13 Exercise 1(a) 27:38 Exercise 3(a) 30:55 Exercise 2(a) #ImproperIntegrals #NumericalIntegration #NumericalAnalysis ━━━━━━━━━━━━━━━━━ MathLens — visual mathematics, carefully explained. Subscribe:    / @lensmath