Answer
$2\ln 2$
Work Step by Step
$A=\displaystyle \int_{a}^{b}f(x)dx$
$A=2\displaystyle \int_{2}^{4}\frac{1}{x\ln x}dx=$
... find an antiderivative (the indefinite integral)
$\left[\begin{array}{l}
u=\ln x\\
du=\frac{1}{x}dx
\end{array}\right]$
$\displaystyle \int\frac{1}{x\ln x}dx=\int\frac{1}{u}du=\ln|u|+C=\ln|\ln x|+C$
$A=2\displaystyle \int_{2}^{4}\frac{1}{x\ln x}dx=2[\ln|\ln x|]_{2}^{4}$
$=2[\ln|\ln 4|-\ln|\ln 2|]\quad$ .. quotient rule...
$=2[\displaystyle \ln(\frac{\ln 4}{\ln 2})]\quad $
... $\ln 4=\ln 2^{2}=2\ln 2$ ...
$A=2\displaystyle \ln(\frac{2\ln 2}{\ln 2})$
$A=2\ln 2$
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