Boredom Buster!

$\begin{aligned} I & = \int_0^\frac \pi 2 \frac {dx}{(4\cos^2 x + 9\sin^2 x)^2} & \small \color{#3D99F6} \text{Since }\sin^2 \theta + \cos^2 \theta = 1 \\ & = \int_0^\frac \pi 2 \frac {dx}{(4 + 5\sin^2 x)^2} & \small \color{#3D99F6} \text{Multiply up and down by }\sec^4 x \\ & = \int_0^\frac \pi 2 \frac {\sec^4 x}{(4\sec^2 x + 5\tan^2 x)^2}dx \\ & = \int_0^\frac \pi 2 \frac {(\tan^2 x + 1)\sec^2 x}{(9\tan^2 x + 4)^2}dx & \small \color{#3D99F6} \text{Let }u = \tan x \implies du = \sec^2 x \ dx \\ & = \int_0^\infty \frac {u^2 + 1}{(9u^2 + 4)^2}du \\ & = \int_0^\infty \frac {\frac 19 (9u^2 + 4)-\frac 49 + 1}{(9u^2 + 4)^2}du \\ & = {\color{#3D99F6} \frac 19 \int_0^\infty \frac {du}{9u^2+4}}+ {\color{#D61F06} \frac 59 \int_0^\infty \frac {du}{(9u^2 + 4)^2}} & \small \color{#3D99F6} \text{Let }v = \frac 32u \implies dv = \frac 32 \ du \\ & = {\color{#3D99F6} \frac 1{54} \int_0^\infty \frac {dv}{v^2+1}}+ {\color{#D61F06} \frac 5{216} \int_0^\frac \pi 2 \frac {d\theta}{\sec^2 \theta}} & \small \color{#D61F06} \text{Let }\tan \theta = \frac 32u \implies \sec^2 \theta \ d\theta = \frac 32 \ du \\ & = \frac 1{54} \tan^{-1} v \bigg|_0^\infty + \frac 5{216} \int_0^\frac \pi 2 \sin^0 \theta \cos^2 \theta \ d\theta \\ & = \frac \pi {108} + \frac 5{432} \color{#3D99F6} B \left(\frac 12, \frac 32\right) & \small \color{#3D99F6} \text{where } B(m,n) \text{ is the beta function.} \\ & = \frac \pi {108} + \frac {5 \color{#3D99F6}\Gamma \left(\frac 12\right)\Gamma \left(\frac 32\right)}{432\color{#3D99F6}\Gamma (2)} & \small \color{#3D99F6} \text{where } \Gamma(s) \text{ is the gamma function.} \\ & = \frac \pi {108} + \frac {5 \color{#3D99F6} \sqrt \pi \left(\frac 12\sqrt \pi \right)}{432\color{#3D99F6}(1!)} \\ & = \frac \pi {108} + \frac {5\pi}{864} \\ & = \boxed{\dfrac {13\pi}{864}} \end{aligned}$

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References:

• Beta function
• Gamma function