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

Explanation

A/Q;-

`int_(0)^(frac(pi)(2)) frac(d x)(2 cos x+4 sin x)`

`=frac{1}{2} int frac{d x}{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}}`

Let `t=tan frac{x}{2} Rightarrow d t=frac{1}{2} sec ^{2} frac{x}{2} d x`

`=frac{1}{2} int_{0}^{frac{pi}{2}} frac{2 d t}{1-t^{2}+4 t}`

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