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For each of the following integrals, give a power or simple exponential function that if integrated on a similar infinite domain will have the same convergence or divergence behavior as the given integral, and use that to predict whether the integral converges or diverges. Note that for this problem we are not formally applying the comparison test; we are simply looking at the behavior of the integrals to build intuition.
(To indicate convergence or divergence, enter one of the words converges or diverges in the appropriate answer blanks.)

$\int_1^{\infty} {1\over e^{2 x} + 1} \,dx$ :	a similar integrand is	so we predict the integral
$\int_1^{\infty} {e^{2 x}\over e^{3 x} + 1} \,dx$ :	a similar integrand is	so we predict the integral
$\int_1^{\infty} {x\over x^2 + 3 x + 1} \,dx$ :	a similar integrand is	so we predict the integral
$\int_1^{\infty} {x^{2} + 1\over x^{3} + 3 x + 1} \,dx$ :	a similar integrand is	so we predict the integral

You can earn partial credit on this problem.