Mathematics - Indefinite Integration Question with Solution | TestHub

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