Question

Calculate the iterated integral.

\(\int {_1^3\int {_1^5} \frac{{Iny}}{{xy}}dydx} \)

Short answer

Calculating the integral we get ,\(\int_{\rm{1}}^{\rm{3}} {\int_{\rm{1}}^{\rm{5}} {\frac{{{\rm{lny}}}}{{{\rm{xy}}}}{\rm{dydx = }}\frac{{{\rm{l}}{{\rm{n}}^{\rm{2}}}\left( {\rm{5}} \right){\rm{ln}}\left( {\rm{3}} \right)}}{{\rm{2}}}} } \)

Step-by-step solution

Step 1:- Given and to find

\(\int_{\rm{1}}^{\rm{3}} {\int_{\rm{1}}^{\rm{5}} {\frac{{{\rm{lny}}}}{{{\rm{xy}}}}{\rm{dydx)}}\left( {{\rm{Given}}} \right)} } \)

The value of the integral) (to find)

Step 2:-Find the value of the integral with respect to y let

\(\begin{array}{l}{\rm{u = ln}}\left( {\rm{y}} \right) & {\rm{so}}\,\,\,\frac{{{\rm{du}}}}{{{\rm{dy}}}}{\rm{ = }}\frac{{\rm{1}}}{{\rm{y}}} & {\rm{du = }}\left( {\frac{{\rm{1}}}{{\rm{y}}}} \right){\rm{dy}}\\{\rm{limit}}\,{\rm{will}}\,{\rm{be}} & {\rm{ln}}\left( {\rm{5}} \right){\rm{iy = 5}}\\\int_{\rm{1}}^{\rm{3}} {\int_{\rm{1}}^{\rm{5}} {\frac{{{\rm{lny}}}}{{{\rm{xy}}}}{\rm{dydx = }}\int_{\rm{1}}^{\rm{3}} {\int_{\rm{1}}^{\rm{5}} {\frac{{\rm{u}}}{{\rm{x}}}{\rm{dudx}}} } } } \\ & {\rm{ = }}\int_{\rm{1}}^{\rm{3}} {\left( {\frac{{\rm{1}}}{{\rm{x}}}\int_{\rm{0}}^{{\rm{ln}}\left( {\rm{5}} \right)} {{\rm{udu}}} } \right){\rm{dx}}} \\ & {\rm{ = }}\int_{\rm{1}}^{\rm{3}} {\left( {\frac{{\rm{1}}}{{\rm{x}}}{{\left( {\frac{{{{\rm{u}}^{\rm{2}}}}}{{\rm{2}}}} \right)}^{{\rm{ln}}\left( {\rm{5}} \right)}}} \right){\rm{dx}}} \\ & {\rm{ = }}\int_{\rm{1}}^{\rm{3}} {\left( {\frac{{\rm{1}}}{{\rm{x}}}\left( {\frac{{{\rm{ln}}{{\left( {\rm{5}} \right)}^{\rm{2}}}}}{{\rm{2}}}{\rm{ - }}\frac{{{{\rm{0}}^{\rm{2}}}}}{{\rm{2}}}} \right)} \right)} {\rm{dx}}\\ & {\rm{ = }}\int_{\rm{1}}^{\rm{3}} {\left( {\frac{{\rm{1}}}{{\rm{x}}}\left( {\frac{{{\rm{l}}{{\rm{n}}^{\rm{2}}}\left( {\rm{5}} \right)}}{{\rm{2}}}} \right)} \right){\rm{dx}}} \\ & {\rm{ = }}\frac{{{\rm{l}}{{\rm{n}}^{\rm{2}}}\left( {\rm{5}} \right)}}{{\rm{2}}}\int_{\rm{1}}^{\rm{3}} {\frac{{\rm{1}}}{{\rm{x}}}{\rm{dx}}} \end{array}\)

Step 3:-Find the value of the final integral

\(\begin{array}{l}\int_{\rm{1}}^{\rm{3}} {\int_{\rm{1}}^{\rm{5}} {\frac{{{\rm{ln}}\left( {\rm{y}} \right)}}{{{\rm{xy}}}}} {\rm{dydx = }}\frac{{{\rm{l}}{{\rm{n}}^{\rm{2}}}}}{{\rm{2}}}\int_{\rm{1}}^{\rm{3}} {\frac{{\rm{1}}}{{\rm{x}}}{\rm{dx}}} } \\ & & {\rm{ = }}\frac{{{\rm{l}}{{\rm{n}}^{\rm{2}}}\left( {\rm{5}} \right)}}{{\rm{2}}}{\left( {{\rm{ln}}\left( {\rm{x}} \right)} \right)^{\rm{3}}}\\ & & {\rm{ = }}\frac{{{\rm{l}}{{\rm{n}}^{\rm{2}}}\left( {\rm{5}} \right)}}{{\rm{2}}}\left( {{\rm{ln}}\left( {\rm{3}} \right){\rm{ - ln}}\left( {\rm{1}} \right)} \right)\\ & & {\rm{ = }}\frac{{{\rm{l}}{{\rm{n}}^{\rm{2}}}\left( {\rm{5}} \right){\rm{ln}}\left( {\rm{3}} \right)}}{{\rm{2}}}\\{\rm{Therfore,}}\int_{\rm{1}}^{\rm{3}} {\int_{\rm{1}}^{\rm{5}} {\frac{{{\rm{ln}}\left( {\rm{y}} \right)}}{{{\rm{xy}}}}{\rm{dydx = }}} } \frac{{{\rm{l}}{{\rm{n}}^{\rm{2}}}\left( {\rm{5}} \right){\rm{ln}}\left( {\rm{3}} \right)}}{{\rm{2}}}\end{array}\)