Home » Blog » Prove some properties of the improper integrals ∫ (sin x) / x and ∫ (cos t) / t2 Prove some properties of the improper integrals ∫ (sin x) / x and ∫ (cos t) / t2 by RoRi March 31, 2016 Prove that the following improper integral converges: Prove that Determine the convergence or divergence of the improper integral Incomplete. Related
for a) use the inequality 0<sin(x)/x<1/cos(x) which is valid for 0<x<pi/4
for b) use the differentiate to check for monotonicity using the fundamental theorem of calculus (don't forget to swap the limits of the integral) and letting I(x) be the integral inside, with substituting u=1/t we can get that I(x)<1/x-1 then using this with the inequality 1-cos(x)<x which is valid for positive x, you can show that xI(x) is decreasing and hence increasing as x decreases to 0, since we have shown that xI(x)<1-x0+ finally you can show that xI(x) is bounded below by (1-xlogx-x) so using the squeeze theorem we get the desired limit.
c) obviously divergent, think about the limit in part (b)