If $\int_0^\pi \frac{d x}{1+2 \sin ^2 x}=k$, then greatest integer less than or equal to $k$ is

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Aicharya · Accepted Answer

$\begin{aligned} & 	ext { } \int_0^\pi \frac{1}{1+2 \sin ^2 x} d x=k \ & 2 \int_0^{\pi / 2} \frac{1}{1+2 \sin ^2 x} d x \ & \quad\left[\because \frac{1}{1+2 \sin ^2 x} 	ext { is an even function }ight] \ & 2 \int_0^{\pi / 2} \frac{\sec ^2 x}{1+	an ^2 x+2 	an ^2 x} d x \ & 2 \int_0^{\pi / 2} \frac{\sec ^2 x}{1+3 	an ^2 x} d x \ & 	ext { Let } 	an x=t \ & \Rightarrow \sec x d x=d t \ & 	ext { When } x ightarrow 0, t ightarrow 0\end{aligned}$ and when $x ightarrow \pi / 2, t ightarrow \infty$ $ \begin{aligned} & 2 \int_0^\pi \frac{1}{1+3 t^2} d t \ \Rightarrow & \frac{2}{\sqrt{3}}\left[	an ^{-1} t \sqrt{3}ight]_0^{\infty} \ \Rightarrow & \frac{2}{\sqrt{3}}\left(\frac{\pi}{2}-0ight) \ \Rightarrow & \frac{\pi}{\sqrt{3}}=\left[\frac{\pi}{\sqrt{3}}ight]=\left[\frac{3.14}{1.73}ight]=1 \end{aligned} $

If $\int_0^\pi \frac{d x}{1+2 \sin ^2 x}=k$, then greatest integer less than or equal to $k$ is

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