I'm having a problem trying to figure out how to integrate the following integral: ln(y)/y with respect to dy with limits of e and 1.

Please can someone assist me

Thanks

\(\displaystyle \int{\frac{\ln{y}}{y}\,dy} = \int{\ln{y}\left(\frac{1}{y}\right)\,dy}\).

Now make the substitution \(\displaystyle u = \ln{y}\) so that \(\displaystyle \frac{du}{dy} = \frac{1}{y}\), the integral becomes

\(\displaystyle \int{u\,\frac{du}{dy}\,dy}\)

\(\displaystyle = \int{u\,du}\)

\(\displaystyle = \frac{1}{2}u^2 + C\)

\(\displaystyle = \frac{1}{2}(\ln{y})^2 + C\).

Therefore:

\(\displaystyle \int_1^e{\frac{\ln{y}}{y}\,dy}= \left[\frac{1}{2}(\ln{y})^2\right]_1^e\)

\(\displaystyle = \left[\frac{1}{2}(\ln{e})^2\right] - \left[\frac{1}{2}(\ln{1})^2\right]\)

\(\displaystyle = \frac{1}{2}\).