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: \(\int_1^3 {\int_1^5 {\frac{{\ln (y)}}{{xy}}} } dydx\)

Let's start with the inside integral.

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

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)

Alternatively, return to the rectangular double integral hub page here:

Double Iterated Integrals Rectangular Regions

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