∫ 4 9 ln ( y ) y d y u = ln ( y ) , d u = 1 y d y d v = 1 y d y , v = 2 y u ⋅ v − ∫ v d u = ( ln ( y ) ) ⋅ ( 2 y ) − ∫ ( 2 y ⋅ 1 y ) d y = 2 y ln ( y ) − ∫ 2 y d y = 2 y ln ( y ) − 2 ∫ 1 y d y = 2 y ln ( y ) − 2 ∫ 1 y = 2 y ln ( y ) − 2 ⋅ 2 y = 2 y ln ( y ) − 4 y = 2 y ( ln ( y ) − 2 ) | 4 9 = ( 2 9 ( ln ( 9 ) − 2 ) ) − ( 2 4 ( ln ( 4 ) − 2 ) ) = ( 6 ln ( 9 ) − 12 ) − ( 4 ln ( 4 ) − 8 ) = ln ( 9 6 ) − 4 + ln ( 4 − 4 ) = ln ( 9 6 ⋅ 4 − 4 ) − 4 = ln ( 9 6 4 4 ) − 4 = 2 ln ( 729 16 ) − 4 = 4 ln ( 27 4 ) − 4 {\displaystyle {\begin{aligned}&\int _{4}^{9}{\frac {\ln(y)}{y}}dy\ \ \ \ \ \ u=\ln(y),du={\frac {1}{y}}dy\ \ \ \ dv={\frac {1}{\sqrt {y}}}dy,v=2{\sqrt {y}}\\[2ex]\\&u\cdot v\ -\int vdu\ =\ {\big (}\ln(y){\big )}\cdot {\big (}2{\sqrt {y}}{\big )}-\int {\big (}2{\sqrt {y}}\ \cdot {\frac {1}{y}}{\big )}dy\\[2ex]&=2{\sqrt {y}}\ln(y)-\int {\frac {2}{\sqrt {y}}}dy\ =\ 2{\sqrt {y}}\ln(y)-2\int {\frac {1}{\sqrt {y}}}dy\ =\ 2{\sqrt {y}}\ln(y)-2\int {\frac {1}{\sqrt {y}}}\\[2ex]&=2{\sqrt {y}}\ln(y)-2\cdot 2{\sqrt {y}}\ =\ 2{\sqrt {y}}\ln(y)-4{\sqrt {y}}\\[2ex]&=2{\sqrt {y}}{\big (}\ln(y)-2){\bigg |}_{4}^{9}\\[2ex]\\&={\bigg (}2{\sqrt {9}}{\big (}\ln(9)-2{\big )}{\bigg )}-{\bigg (}2{\sqrt {4}}{\big (}\ln(4)-2{\big )}{\bigg )}\ =\ {\bigg (}6\ln(9)-12{\bigg )}-{\bigg (}4\ln(4)-8{\bigg )}\\[2ex]&=\ln {\big (}9^{6}{\big )}-4+\ln {\big (}4^{-4}{\big )}\ =\ \ln {\big (}9^{6}\cdot 4^{-4}{\big )}-4\ =\ \ln {\bigg (}{\frac {9^{6}}{4^{4}}}{\bigg )}-4\ =\ 2\ln {\bigg (}{\frac {729}{16}}{\bigg )}-4\\[2ex]&=4\ln {\bigg (}{\frac {27}{4}}{\bigg )}-4\end{aligned}}}
![{\displaystyle {\begin{aligned}&\int _{4}^{9}{\frac {\ln(y)}{y}}dy\ \ \ \ \ \ u=\ln(y),du={\frac {1}{y}}dy\ \ \ \ dv={\frac {1}{\sqrt {y}}}dy,v=2{\sqrt {y}}\\[2ex]\\&u\cdot v\ -\int vdu\ =\ {\big (}\ln(y){\big )}\cdot {\big (}2{\sqrt {y}}{\big )}-\int {\big (}2{\sqrt {y}}\ \cdot {\frac {1}{y}}{\big )}dy\\[2ex]&=2{\sqrt {y}}\ln(y)-\int {\frac {2}{\sqrt {y}}}dy\ =\ 2{\sqrt {y}}\ln(y)-2\int {\frac {1}{\sqrt {y}}}dy\ =\ 2{\sqrt {y}}\ln(y)-2\int {\frac {1}{\sqrt {y}}}\\[2ex]&=2{\sqrt {y}}\ln(y)-2\cdot 2{\sqrt {y}}\ =\ 2{\sqrt {y}}\ln(y)-4{\sqrt {y}}\\[2ex]&=2{\sqrt {y}}{\big (}\ln(y)-2){\bigg |}_{4}^{9}\\[2ex]\\&={\bigg (}2{\sqrt {9}}{\big (}\ln(9)-2{\big )}{\bigg )}-{\bigg (}2{\sqrt {4}}{\big (}\ln(4)-2{\big )}{\bigg )}\ =\ {\bigg (}6\ln(9)-12{\bigg )}-{\bigg (}4\ln(4)-8{\bigg )}\\[2ex]&=\ln {\big (}9^{6}{\big )}-4+\ln {\big (}4^{-4}{\big )}\ =\ \ln {\big (}9^{6}\cdot 4^{-4}{\big )}-4\ =\ \ln {\bigg (}{\frac {9^{6}}{4^{4}}}{\bigg )}-4\ =\ 2\ln {\bigg (}{\frac {729}{16}}{\bigg )}-4\\[2ex]&=4\ln {\bigg (}{\frac {27}{4}}{\bigg )}-4\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/12560f097d9377ae996e58393b65d0691f1c1c8a)