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How do you evaluate the integral #int 1/x dx# from 1 to #oo# if it converges?

Answer 1

Isaiah Allen

The integral does not converge.

#int_1^a 1/x dx=lna-ln1=lna#

We know that #lnx# is a strict monotonic increasing function so

#lim_(a->oo)lna=oo#. So we conclude that

#lim_(a->oo)int_1^a 1/x dx = oo# so does not converge.

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Answer 2

Charles Anctil

To evaluate the integral ∫(1/x) dx from 1 to ∞ if it converges, we use the limit definition of definite integrals. The integral converges if the limit of the integral as the upper limit approaches infinity exists.

∫(1/x) dx from 1 to ∞ = lim┬(b→∞)⁡〖∫_1^b (1/x) dx〗

= lim┬(b→∞)⁡[ln|x|]┬1^b

= lim┬(b→∞)⁡[ln|b| - ln|1|]

= lim┬(b→∞)⁡[ln|b|]

= ∞

Since the limit of the integral as the upper limit approaches infinity is ∞, the integral diverges.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.