Mathematics - Definite Integration Question with Solution | TestHub

$f\left(x\right)= \underset{1}{\overset{x}{\int }}\frac{\log t}{1+t} dt$$\Rightarrow f\left(x\right)=\underset{1}{\overset{x}{\int }}\frac{\mathit{log}z}{(1+z)}dz$
And$f\left(\frac{1}{x}\right)= \underset{1}{\overset{\frac{1}{x}}{\int }}\frac{\log t}{1+t} dt$
Put$t=\frac{1}{z}$
$dt= -\frac{1}{z^2} dz$
$f\left(x\right)= \underset{1}{\overset{x}{\int }}\frac{\log z}{z^2\left(1+ \frac{1}{z}\right)} \cdot dz$
$f\left(x\right)= \underset{1}{\overset{x}{\int }}\frac{\log z}{z(1+z)} dz$
$f\left(x\right)+f\left(\frac{1}{x}\right)= \underset{1}{\overset{x}{\int }}\log z \left[\frac{1}{1+z}+\frac{1}{z(1+z)}\right] dz$
$= \underset{1}{\overset{x}{\int }}\frac{1}{z}\log z dz$
Put$\log z=P$$\Rightarrow \frac{1}{z} dz=dP$
$\underset{1}{\overset{x}{\int }}P\cdot dP={\left(\frac{P^2}{2}\right)}_1^x=\frac{1}{2} {\left({\log }^{2}z\right)}_1^x= \frac{{\left(\log x\right)}^2}{2}$