calculus

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how do you solve the integral of 1/[(square root of x)(lnx)] from 2 to infinity?
i did the p- integral theorem with 1/square root of x and got it to be a divergent integral. however i was told this was the wrong way and that i should do it by integration by parts. but i can't figure it out by that method. please help. thanks.

  • calculus -

    You can proceed as follows. Substitute:

    x = e^t. Then the integral becomes:

    Integral from ln(2) to infinity of

    e^(t/2)/t dt

    We can get rid of the factor 2 in the exponential by putting y = t/2. The integral becomes:

    Integral from 1/2 ln(2) to infinity of

    e^(y)/y dy

    For positive y we have

    e^(y) > 1

    It follows from this that the integral is larger than

    Integral from 1/2 ln(2) to infinity of

    1/y dy

    but this is already divergent, so the integral diverges.

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