prob6sol - Math 151 Spring 2008 Problem Set # 6 Improper...

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Math 151 Spring 2008Problem Set # 6Improper Integrals: Part 2Solutions1.Ifx >00cos2(4x)x3/21x3/2.Since3/2>1, the improper integralZπ1x3/2dxconverges. By the comparison theorem,Zπcos2(4x)x3/2dxconverges. Thus,Zπcos2(4x)x3/2dxconverges as well.2.Ifx1thenx1so that0<exxex.The improper integralR1exdxconverges. Indeed,Z1exdx=limb+Zb1exdx=limb+³ex¯¯b1´=limb+¡eb+e1¢=e1.By the comparison test, the given integral converges as well.3.Ifx9thenx3so thatxx+ 43x+ 4>0.The improper integralZ93x+ 4dxdiverges. Indeed,Zb93x+ 4dx= 3 ln (|x+ 4|)|b9= 3 ln (b+ 9)3 ln (13),so thatlimb→∞Zb93x+ 4dx= limb→∞(3 ln (b+ 9)3 ln (13)) = +.By the comparison test, the given integral diverges as well.1
4.Ifx1then0<1x3+ 2x+ 9<1x3,and the integralZ11x3dx

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Term
Winter
Professor
Geveci
Tags
Improper Integrals, Integrals, lim, dx, Riemann integral, Henstock Kurzweil integral

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