Question

how do you determine the convergence of :

definite integral from 1--> infinity of lnx/(x^3)?

i set the problem as
lim (R--->infinity) of the integral of lnx/(x^3) from 1--->R, but i can't compute the integral.

Answer 1

something *( x^-2) ln x - something else*x^something call it k

say

a (x^-2) ln x - b x^c

differentiate to get
a (x^-2)(x^-1) + a [ ln x] (-2) x^-3 + b c x^(c-1)

clearly -2 a = 1
so
a = -1/2

so - (1/2) x^-3 = b c x^(c-1)
so
c-1 = -3
c = -2
but b c = - a = 1/2
b = -1/4
a (x^-2) ln x - b x^c
is
-(1/2)[ln x]/x^2 +(1/4) x^(1/2)
check my arithmetic!!!
so in the end

Answer 2

-(1/2)[ln x]/x^2 +1/(4x^2)

Answer 3

wait, are you doing the convergent problem or the partial fraction problem?

Answer 4

I am just doing the integral of (ln x)dx / x^3

Answer 5

In other words I think
integral of ln x dx/x^3= (-1/2)(ln x)/x^2 + 1/(4x^2) + constant

Answer 6

but check my arithmetic !!