Evaluate the following integral: `int_0^(pi//2)1/(2cos x+4sinx)dx`

Explanation

`int_{0}^{pi} frac{d x}{2 cos x+4 sin x}`
`
=frac{1}{2} int frac{{dx}}{cos {x}+2 sin {x}}`
`
=frac{1}{2} int_{0}^{frac{pi}{2}} frac{d x}{(frac{1-tan ^{2} frac{x}{2}}{1+tan ^{2} frac{x}{2}})+2(frac{2 tan frac{x}{2}}{1+tan ^{2} frac{x}{2}})}`
`
=frac{1}{2} int_{0}^{frac{pi}{2}} frac{(1+tan ^{2} frac{{x}}{2}) d {x}}{1-tan ^{2} frac{{x}}{2}+4 tan frac{{x}}{2}}`
`
=frac{1}{2} int_{0}^{frac{pi}{2}} frac{sec ^{2} frac{{x}}{2} d x}{1-tan ^{2} frac{{x}}{2}+4 tan frac{{X}}{2}}`

Suppose `t=tan frac{{x}}{2} Rightarrow {dt}=frac{1}{2} sec ^{2} frac{{x}}{2} {dx}`
`
=frac{1}{2} int_{0}^{frac{pi}{2}} frac{2 {dt}}{1-{t}^{2}+4 {t}}`
`
=-int_{0}^{frac{-}{2}} frac{{dt}}{{t}^{2}-4 {t}-1}`
`
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