Question

\[\int\limits_0^\pi \frac{1}{3 + 2 \sin x + \cos x} dx\]

Answer 1

\[Let\ I = \int_0^\pi \frac{1}{3 + 2 \sin x + \cos x} d x . Then, \]
\[I = \int_0^\pi \frac{1}{3 + 2\left( \frac{2 \tan\frac{x}{2}}{1 + \tan^2 \frac{x}{2}} \right) + \frac{1 - \tan^2 \frac{x}{2}}{1 + \tan^2 \frac{x}{2}}} d x\]
\[ \Rightarrow I = \int_0^\pi \frac{1 + \tan^2 \frac{x}{2}}{2 \tan^2 \frac{x}{2} + 4 \tan \frac{x}{2} + 4} dx\]
\[Let \tan \frac{x}{2} = t . Then, \frac{1}{2} \sec^2 \frac{x}{2} dx = dt\]
\[When\ x = 0, t = 0\ and\ x = \pi, t = \infty \]
\[ \therefore I = \int_0^\infty \frac{2 dt}{2 t^2 + 4t + 4}\]
\[ \Rightarrow I = \int_0^\infty \frac{dt}{\left( t + 1 \right)^2 + 1}\]
\[ \Rightarrow I = \left[ \tan^{- 1} \left( t + 1 \right) \right]_0^\infty \]
\[ \Rightarrow I = \frac{\pi}{2} - \frac{\pi}{4}\]
\[ \Rightarrow I = \frac{\pi}{4}\]