Question

If I = $\frac{dx}{x^{2}}$, then I equals

• A. 2
• B. 2/e
• C. 2( 1 – 1/e)
• D. 0

Answer 1

Answer: (C)

2( 1 – 1/e)

Answer 2

Put $\frac{1}{x}$= t, so that

I = $\int_{e}^{1/e}\left| \log\frac{1}{t} \right|$ (–1) dt

= $\int_{1/e}^{e}\left| - \log t \right|dt$

= $\int_{1/e}^{1}{( - \log t)dt}$+ $\int_{1}^{e}{(\log t)dt}$

= (–t log t + t )]$\rbrack_{1/e}^{1}$+ (t + log t –t) $\rbrack_{1}^{e}$

= 1 – $\frac{1}{e}$– $\frac{1}{e}$+ e – e + 1

= 2(1 – 1/e)