Determine wether the improper integral converges or diverges. Evaluate if it converges:

Determine wether the improper integral converges or diverges. Evaluate if it converges:
∞
∫ xe^(x/3) dx
°

This can be done directly via integration by parts, but it is unnecessary.
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This diverges automatically, because lim(x→∞) xe^(x/3) = ∞, which is nonzero.

Alternately, note that for all non-negative x, we have xe^(x/3) ≥ x.
Since ∫(x = 0 to ∞) x dx = (1/2)x^2 {for x = 0 to ∞} = ∞, we conclude that the improper integral in question also diverges by the Comparison Test.

I hope this helps!