Definite Integration
King's Rule — Sixth Root of Tan
nta_pyq_2026_jan
Grade 12
Question:
The value of the integral $\displaystyle\int_{\pi/24}^{5\pi/24}\frac{dx}{1+\sqrt[6]{\tan 2x}}$ is:
$dfrac{pi}{3}$
$dfrac{pi}{12}$
$dfrac{pi}{18}$
$dfrac{pi}{6}$
Step-by-Step Solution
Key Concept: Apply King's rule $x\to\dfrac{\pi}{4}-x$: $I=\int_{\pi/24}^{5\pi/24}\dfrac{\sqrt[6]{\tan 2x}}{1+\sqrt[6]{\tan 2x}}\,dx$. Adding: $2I=\int_{\pi/24}^{5\pi/24}dx=\dfrac{5\pi}{24}-\dfrac{\pi}{24}=\dfrac{\pi}{6}$.
$I=\pi/12$.
Correct Answer: 2