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

Explanation

`I=int_{0}^{pi / 2} frac{d n}{4 cos n+2 sin n}`

`I=int_{-}^{pi / 2} frac{d n}{frac{4-4 tan ^{2} 2_{4}}{}+frac{4 tan ^{2} 3_{2}}{1+tan ^{2} n_{2}}} frac{1+tan ^{2} n_{2}}{1}`


`I=int frac{2 d t}{t-4 t^{2}+4 t}`

`I=-frac{1}{2} int frac{d t}{t^{2}-t-1}=-frac{1}{2} int frac{d t}{(t-1 / 2)^{2}-(sqrt{7 / 2})^{2}}`

`I=-frac{1}{2} cdot frac{1}{2 cdot sqrt{2}} cdot log |frac{t-1 / 2-sqrt{tau} / 2}{t-1 / 2+sqrt{r} / 2 cdot 1}`

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`I=-frac{1}{2 sqrt{5}}[log |frac{tan ^{2} / 2-(frac{1+sqrt{5}}{2})}{tan ^{2} 2-(frac{1-sqrt{5}}{2})}|]_{0}^{frac{2}{2}}`


`I=-frac{1}{2 sqrt{5}}(log |frac{1-frac{1}{ 2}-sqrt{sqrt{2}}}{1-frac{1}{ 2}+sqrt{2}}|-log |frac{1+sqrt{5}}{1-sqrt{5}}|)`