How do you integrate lnx / x?

1 Answer
Apr 30, 2016

Use a u-substitution to get (lnx)^2/2+C.

Explanation:

At first glance, this integral looks a little confusing because we have a function divided by another function (and those tend to be difficult to work with). But, after rewriting intlnx/xdx as int1/xlnxdx, we can see something interesting: we have lnx and its derivative, 1/x, in the same integral, making it a textbook case of a u-substitution:
Let color(blue)u=color(blue)lnx->(du)/dx=1/x->color(red)(du)=color(red)(1/xdx)

Thus, the integral intcolor(red)(1/x)color(blue)(lnx)color(red)(dx) becomes:
intcolor(blue)(u)color(red)(du)

Now, isn't this much easier? Using the reverse power rule, the integral evaluates to u^2/2+C. Because u=lnx, we can say:
intlnx/xdx=(lnx)^2/2+C