JEE Main 2026 · 8 Apr · Shift 2, Mathematics question 22

Identity. With $A=x+\tfrac\pi3$, $B=x-\tfrac\pi3$ and $A-B=\tfrac{2\pi}3$:
$$\cot A\cot B+1=\cot(A-B)(\cot B-\cot A)=-\frac1{\sqrt3}\left(\cot B-\cot A\right).$$

Integrate.
$$I=-\frac1{\sqrt3}\left[\ln\left|\frac{\sin(x-\pi/3)}{\sin(x+\pi/3)}\right|\right]_{\pi/6}^{\pi/4}=-\frac1{\sqrt3}\left[\ln(2-\sqrt3)-\ln\tfrac12\right]=-\frac1{\sqrt3}\ln\left(4-2\sqrt3\right).$$

Value. $4-2\sqrt3=(\sqrt3-1)^2$, so $I=-\dfrac2{\sqrt3}\ln(\sqrt3-1)$, $\alpha=-\dfrac2{\sqrt3}$ and $9\alpha^2=12$.

Answer: 12