لتنشيط الذهن

$\LARGE {\color{DarkRed} (1)\int \frac{2x}{(1+x)^{2}}dx=2\int \frac{x+1-1}{(x+1)^{2}}dx}$
${\color{DarkRed} =2[\int \frac{1}{(x+1)}dx-\int \frac{1}{(x+1)^{2}}dx]}$
${\color{DarkRed} =2[ln(x+1)+\frac{1}{x+1}]+c}$

[SIZE=4]$\LARGE {\color{Cyan} (2)\int \frac{4}{xln3x}dx ,,,put,y=ln3x\rightarrow dy=\frac{1}{x}dx}$[/SIZE]
${\color{Cyan} \Rightarrow I=4\int \frac{1}{y}dy=4lny+c=4lnln3x+c}$

$\LARGE {\color{Red} (3)\int \frac{(1+lnx)^{2}}{x}dx=\frac{(1+lnx)^{3}}{3}+c,,} $