JEE Main 2024 — Indefinite Integration Question with Solution

$\begin{aligned} & \int \frac{\sec ^2 x d x}{a^2 \tan ^2 x+b^2} \\ & \text { let } \tan x=t \\ & \sec ^2 d x=d t \\ & \int \frac{d t}{a^2 t^2+b^2} \\ & \frac{1}{a^2} \int \frac{d t}{t^2+\left(\frac{b}{a}\right)^2} \\ & \frac{1}{a^2} \frac{1}{\frac{b}{a}} \tan ^{-1}\left(\frac{t}{b} a\right)+c \\ & \frac{1}{a b} \tan ^{-1}\left(\frac{\alpha}{b} \tan x\right)+c \end{aligned}$ on comparing $\frac{\mathrm{a}}{\mathrm{b}}=3$ $\begin{aligned} & a b=12 \\ & a=6, b=2 \end{aligned}$ maximum value of $6\sin x+2\cos x\text{ is }\sqrt{40}$