intergrate the following?

Question

1/xlnx by substitution

Answer 1

let u=lnx
then du = 1/x dx..

use substitution, and walla!!!... you get answer..

I'll let you finish rest..

its pretty easy from here.

Answer 2

I am assuming that ln x is also at the denominator.

Usually, it'll be a big clue to know that your function consist of a product of 1/x and 1/ln x. So, what we can do is to find the derivative of one of these two functions, and see if we can get the other one as the derivative. In this case, it doesnt matter if the one that you're trying to differentiate is at the numerator or denominator, as long as you can get the exact derivative (in this case, it does matter if it is at the numerator of denominator)

Ok, back to the problem..... you can see that by differentiating ln x. you get 1/x (which means by differentiating one of the products, you get the other one as its derivative)

so, the substitution will be

u = ln x du = 1/x dx

so, back to your original function, we substitute with all the values we get (int means integrate)

int 1/(x lnx) dx = int 1/(x*u) dx = int 1/u du = ln u

To get beck to x, substitute back u = ln x into the equation, and you get the final answer

ln u = ln (ln x)

Hope this helps

Answer 3

Int (1/x) (1/log x) dx

Put log x = t

(1/x) dx = dt

Int (1/t) dt

log t + c

log (log x) + c

Answer 4

u=ln x
du=1/x dx
therefore = integral (1/u) du

= ln u
= ln ln x

Answer 5

what?