Explanation:
(A) : Given,
\(I=\int_{1}^{e^{2}} \frac{d x}{x(1+\log x)^{2}}\)
Let, \(1+\log \mathrm{x}=\mathrm{t}\)
\(\mathrm{dt}=\frac{1}{\mathrm{x}} \mathrm{dx}\)
For limit,
When, \(\mathrm{x}=1\), then \(\mathrm{t}=1\)
When, \(x=e^{2}\) then, \(t=3\)
\(\therefore \quad \mathrm{I}=\int_{1}^{3} \frac{\mathrm{dt}}{(\mathrm{t})^{2}}=\int_{1}^{3} \mathrm{t}^{-2} \mathrm{dt}=\left[-\frac{1}{\mathrm{t}}\right]_{1}^{3}=-\frac{1}{3}+\frac{1}{1}=\frac{2}{3}\)
