Let $$ \int_{0}^{ \infty} \frac{x+ \sin (x)}{1+x^2} dx, \ \int_{0}^{1} \frac{ \cos (x) }{ x^{ 3/2 } } dx, \ \int_{0}^{ \infty } \frac{ 1}{1+x} dx $$ be improper integrals we want to study.

I want to use comparison test to determine whether the following improper integrals are divergent or convergent. I found it easier to determine the convergent function, but have trouble dealing with divergent functions. Also, for normal integrals like second one, what is a good strategy to tell if it is convergent or divergent?


For (1) you only need to check in the interval $\;[1,\infty)\;$ (can you see why?) , and here

$$\frac{x+\sin x}{1+x^2}\ge\frac{x-1}{2x^2}$$

The same applies for the third case.

As for the second one on the given interval

$$\frac{\cos x}{x^{3/2}}\ge\frac{\cos1}{x^{3/2}}$$

  • $\begingroup$ I know I can start from 1, but I never understand why..... $\endgroup$ – XXWANGL Feb 22 '16 at 23:36
  • $\begingroup$ I think I can start from 1 only when the integral exists in the interval [0,1] ? $\endgroup$ – XXWANGL Feb 22 '16 at 23:38
  • $\begingroup$ @XXWANGL Exactly. In both cases 1-3 the integral is a "normal" Riemann integral of a continuous function, so it certainly exists. $\endgroup$ – DonAntonio Feb 22 '16 at 23:43
  • $\begingroup$ It makes sense! Thank u so much! $\endgroup$ – XXWANGL Feb 22 '16 at 23:44

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