Question

Help on number 2 A-C

Math 166 Spring 2020 Lab 12 - Integration Strategies and Improper Integrals 1. Evaluate the following integrals. (a) | In(x2 + 2a) dx 100 dx (8) Jo Je to (1) ["* sin(a) Vsee(2) de 5 1 11 x² – 2x – 3 dx 87/2 13 x(lnx)2 de (c) / tarda (1) [4x*e*** de 2. For what values of p do the following improper integrals converge? (1/2 da (0) Le 2 In () Jo 3. Give an example of (a) A function f(x), continuous for x > 1, such that lim f(x) = 0, but f(x) dx diverges. (b) A function f(x), continuous at x = 2 and x = 5, such that the integral dc is improper and divergent. 4. Recall that the doubly infinite integral / f(x) dx converges if and only if there exists a real number a such that both improper integrals L. f() dx and so f(x) dx converge. In this case, Lºs(adr = ["$(x)de + Sºs(e)dir. -00 (a) Show that dx diverges. (b) Show that limt-00 / x dx = 0. J-t (c) Explain the exact reason why /s(a) dx + lim00 ["s(z) du? (d) Suppose that f is a continuous odd function. Prove that if Sof(x) dx converges, then dx=0. (e) Suppose that f is a continuous even function. Prove that if f f() dx converges, then dx = 2 8°° (x)dx. 5. Describe the contributions of each of your group members to Labs 11 and 12.

Answer 1

2. For what values of p do the following improper integrals converge? 1/2 dx - dr Apply Integration By Parts: u= ln(x), We have slage) et = -x P+(x) – 5xl-p+ajat So, we have Singap) dx = -****+p+31(x) = x+P++C for Phi, gelapor los ex is ao as x oo & for 871, 294 / as amo. Hence ko lncx) is convergent if ?>i.. eta de Here, we have Sep*a = Sem diu Apply u- substitution: u=px 00 (b) So, we have S px de = 1DX + C , Now, for P>o, e os as xto and for pso we have either eff=1 (if P=0) & ep*o (ifp <1), 2 x o... So, interferol sou epx dx is convergent if pso. (1/2 da (c) J. 2 [[n(7) We have Seconds)odt = sa zdu Apply u – substitution: 4 = \n(x) = S u-P du -p+1 Apply the Power Rule: JxCar= a + -1 -P + 1 so. Stader = 1n 77(0)+c Now, I'm dx = {(lns (2) / 11]" But, this is ondefined infinite for all values of P. (P)=0 makes denominator zero & Inox) is Endefined for x=o. Hence, the firen definite integral is direnfent. So, it is not convergent for any value of P.