Question

integral of ((lny)/y )

Answer 1

To find the integral of \(\frac{\ln y}{y}\), we can use the substitution method. Let's set:

\[
u = \ln y
\]

Then, we find the differential of \(u\):

\[
du = \frac{1}{y} dy \quad \text{or} \quad dy = y \, du = e^u \, du
\]

Substituting \(y = e^u\) into the integral, we have:

\[
\int \frac{\ln y}{y} \, dy = \int u \, du
\]

Now, we can integrate \(u\):

\[
\int u \, du = \frac{u^2}{2} + C
\]

Substituting back \(u = \ln y\):

\[
\int \frac{\ln y}{y} \, dy = \frac{(\ln y)^2}{2} + C
\]

Thus, the integral of \(\frac{\ln y}{y}\) is:

\[
\int \frac{\ln y}{y} \, dy = \frac{(\ln y)^2}{2} + C
\]

where \(C\) is the constant of integration.