The Natural Logarithm Function?

Question

integrate from 1<=x<=2
( 2 ln (X) d(x)) / x

answer = (ln2)^2

i cant get the answer can you show the steps how to do this thx

Answer 1

Rewrite the integral like this:

ln x (2/x) dx = (ln x) 2 [(1/x) dx]

Now, let u = ln x. Then du = [(1/x) dx]. We have 2 [(1/x) dx], which is 2 du.

We can see then that this integral is in the form:

ƒ u (2 du) = 2 ƒ u du = 2 (u²/2) = u² | evaluated from 1 to 2.

Now all we have to do is put (ln x) back in for u and evaluate the integral at the upper and lower bounds for (ln x):

(ln 2)² - (ln 1)² = (ln 2)² - 0² = (ln 2)² - 0 = (ln 2)² (your final result).

Hope this helps you out.