### Examples of Double Iterated Integrals over Rectangular Regions (Example 2)

These kinds of double integrals have numbers as the integral boundaries for the x and y variables. Calculate the iterated integral: $$\int_1^3 {\int_1^5 {\frac{{\ln (y)}}{{xy}}} } dydx$$

$\int_1^5 {\frac{{\ln (y)}}{{xy}}dy}$

So we integrate this with respect to y. We can just use regular substitution:

$u = \ln (y)$

$du = \frac{1}{y}dy$

$u(1) = \ln (1) = 0$

$u(4) = \ln (5)$

Substituting and integrating:

$\frac{1}{x}\int_0^{\ln (5)} {udu}$

$\frac{1}{x}\left[ {\frac{{{{\left[ {\ln (5)} \right]}^2}}}{2} - \frac{{{{(0)}^2}}}{2}} \right]$

$\frac{{{{\left[ {\ln (5)} \right]}^2}}}{{2x}}$

That's the inner integral, and we just plug it back into the original. Let's call the original integral "I".

$I = \int_1^3 {\frac{{{{\left[ {\ln (5)} \right]}^2}}}{{2x}}dx}$

$I = \frac{{{{\left[ {\ln (5)} \right]}^2}}}{2}\int_1^3 {\frac{1}{x}} dx$

$I = \frac{{{{\left[ {\ln (5)} \right]}^2}}}{2} \cdot \left[ {\ln (3) - \ln (1)} \right]$

$I = \frac{{{{\left[ {\ln (5)} \right]}^2}\ln (3)}}{2}$

Please see our other rectangular double integral examples below:

Double Iterated Integrals Rectangular Region (Ex1)