Q. $\int_{1}^{x}\frac{\log{(x^{2})}}{x}\ dx$ is equal to
A) $({\log{x)}}^{2}$
B) $\frac{1}{2}({\log{x)}}^{2}$
C) $\frac{\log x^{2}}{2}$
D) None of these
Explanation
(a)
Let $I = \int_{1}^{x}{\frac{\log{(x^{2})}}{x}dx = 2\int_{1}^{x}{\log{x.\frac{1}{x}dx}}}$
Put $\log{x = t\ \Rightarrow \ \frac{1}{x}dx = dt}$
$\therefore I = 2\int_{0}^{\log x}{t\ dt = 2\left\lbrack \frac{t^{2}}{2} \right\rbrack_{0}^{\log{x\ }} =}\left( \log x \right)^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.