Question

Integrate

$\int \frac{3+4\sin x+2\cos x}{3+2\sin x+\cos x}dx$

Answer 1

$2x-3\int \frac{dt}{t^2+2\times \frac{1}{4}x+\frac{1}{16}+2}+c$

$=2x-3\int \frac{dt}{{(t+\frac{1}{4})}^2+\frac{31}{16}}+c$

$=2x-3\int \frac{dt}{{(t+\frac{1}{4})}^2+{\left(\frac{\sqrt{31}}{4}\right)}^2}+c$

$=2x-3\frac{1}{\frac{\sqrt{31}}{4}}{\tan }^{-1}\frac{t+\frac{1}{4}}{\frac{\sqrt{31}}{4}}+c$

$=2x-\frac{12}{\sqrt{31}}{\tan }^{-1}{\tan }^{-1}\frac{4t+1}{\sqrt{31}}+c$

$=2x-\frac{12}{\sqrt{31}}{\tan }^{-1}\left(\frac{4\tan \frac{x}{2}+1}{\sqrt{31}}\right)+c$

Hence, the answer is $2x-\frac{12}{\sqrt{31}}{\tan }^{-1}\left(\frac{4\tan \frac{x}{2}+1}{\sqrt{31}}\right)+c.$