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Notes 10 INT (Improper Integrals)

# Notes 10 INT (Improper Integrals) - MIT OpenCourseWare...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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INT. IMPROPER INTEGRALS In deciding whether an improper integral converges or diverges, it is often awkward or impossible to try to decide this by actually carrying out the integration, i.e., finding an antiderivative explicitly. For example both of these two improper integrals converge: 0" daz z+3x s +22 + 1' and 1 z but there is no explicit antiderivative for the second integral, and finding one for the first would be a hairy exercise in partial fractions, even ifone were able to factor the denominator. Instead of explicit integration, therefore, we show they converge by using estimation instead, comparing them with simpler integrals which are known to converge. Thus, for the first, 1 1 s6 + 3Zs + 2 +1 - ~' X> 0, so that J d0 1 " dsz + < z*=E. x - z+3X3+2X2+1 Since the right hand integral converges, so does the left, which is smaller (but still positive). In a similar way, for the second integral, we estimate 1 1 ~P,'FF 5 X> 0, so that I F v& d < <i F•/- = 2. In the same way we can show the divergence say of
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Notes 10 INT (Improper Integrals) - MIT OpenCourseWare...

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