Evaluate: `int_0^(pi//2)1/(4sin^2x+5cos^2x)dx`

Explanation

`int_{0}^{frac{pi}{2}} frac{d x}{4 sin ^{2} x+5 cos ^{2} x}`
`
=int_{0}^{pi / 2}(frac{sec ^{2} x d x}{4 tan ^{2} x+5})`
`
=int_{0}^{infty} frac{d t}{4 t^{2}+5}`

We have ` t=tanx`
`
=frac{1}{4} int_{0}^{infty} frac{d t}{t^{2}+(frac{sqrt{5}}{2})^{2}}`
`
=frac{1}{4} times frac{2}{sqrt{5}} tan ^{-1}(frac{2 t}{sqrt{5}})_{0}^{infty}`
`
=(1 / 2 sqrt{5})((pi / 2)-0) `
`
=pi / 4 sqrt{5}`