Definite Integration
Symmetric Integral — arctan
nta_pyq_2023_jan
Grade 12
Question:
The value of the integral $\displaystyle\int_{1/2}^{2}\dfrac{\tan^{-1}x}{x}\,dx$ is equal to:
$pilog_e 2$
$dfrac{1}{2}log_e 2$
$dfrac{pi}{4}log_e 2$
$dfrac{pi}{2}log_e 2$
Step-by-Step Solution
Key Concept: Let $I=\int_{1/2}^2\frac{\tan^{-1}x}{x}dx$. Sub $x\to1/x$: $I=\int_{1/2}^2\frac{\tan^{-1}(1/x)}{x}dx$. Adding: $2I=\frac{\pi}{2}\int_{1/2}^2\frac{dx}{x}$.
$\dfrac{\pi}{2}\log_e 2$.
Correct Answer: 4