Q. The value of $\int_{1}^{e^{2}}\frac{dx}{x{(1 + \log{x)}}^{2}}$ is
A) $\frac{2}{3}$
B) $\frac{1}{3}$
C) $\frac{3}{2}$
D) $\ln 2$
Explanation
(a)
Let $I = \int_{1}^{e^{2}}\frac{dx}{x(1 + \log{x)\hat{}2\ }}$
Put$\ 1 + \log{x = t\ \Rightarrow \ \frac{1}{x}dx = dt}$
$\therefore I = \int_{1}^{3}{\frac{1}{t^{2}}dt = \left\lbrack - \frac{1}{t} \right\rbrack_{1}^{3}} = \frac{2}{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.