# 2017 SE101 Lecture 07(1).pdf - 2017 SE101 Differential...

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2017 SE101 Differential Equations March 23, 2017 7 Applications to improper in- tegrals Theorem 7.1. Let f, g : [ a, ) R be continuous function satisfying f ( x ) g ( x ) 0 for all x a . Then Z a g ( x ) dx converges if Z a f ( x ) dx does. Z a f ( x ) dx diverges if Z a g ( x ) dx does. Example 7.2. The improper integral Z 0 e - x 2 dx converges. First, we note that Z 0 e - x 2 dx = Z 1 0 e - x 2 dx + Z 1 e - x 2 dx Since the first term is a definite integral, the conver- gence depends only on the convergence of Z 1 e - x 2 . Note that for x 1, e - x 2 e - x . Since Z 1 e - x dx = lim t →∞ Z t 1 e - x dx = lim t 1 - 1 e - t = 1 e , the improper integral Z 1 e - x 2 also converges. Example 7.3. Does Z 1 0 log xdx converges? By sub- stitute t = 1 /x , we note that Z 1 0 log xdx = Z 1 log 1 t ( - t 2 ) dt = - Z 1 - log t - t 2 dt = - Z 1 log t t 2 dt Thus the convergence of the improper integral de- pends on the convergence of Z 1 log t t 2 dt . We also note that log t t for all t 1, and Z 1 t t 2 dt = Z 1 1

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