Cho \( F\left( x \right) = \smallint \frac{{\ln x}}{{x\sqrt {1 – \ln x} }}dx\) , biết F( e ) = 3 , tìm F( x ) = ? 

\(
F\left( x \right) = \smallint \frac{{\ln x}}{{x\sqrt {1 – \ln x} }}dx\)

Đặt: 

\(
\sqrt {1 – \ln x} = t \Rightarrow 1 – \ln x = {t^2} \Rightarrow \ln x = 1 – {t^2} \Rightarrow \frac{1}{x}dx = – 2tdt\)

\(\begin{array}{l}
\Rightarrow F\left( x \right) = \smallint \frac{{1 – {t^2}}}{t}\left( { – 2tdt} \right) = – 2\smallint \left( {1 – {t^2}} \right)dt\\
= – 2t + \frac{2}{3}{t^3} + C = – 2\sqrt {1 – \ln x} + \frac{2}{3}\left( {1 – \ln x} \right)\sqrt {1 – \ln x} + C\\
\begin{array}{*{20}{l}}
{F\left( e \right) = – 2\sqrt {1 – 1} + \frac{2}{3}\left( {1 – 1} \right)\sqrt {1 – 1} + C = 3 \Rightarrow C = 3}\\
{ \Rightarrow F\left( x \right) = – 2\sqrt {1 – \ln x} + \frac{2}{3}\left( {1 – \ln x} \right)\sqrt {1 – \ln x} + 3}
\end{array}
\end{array}\)