How to calculate antiderivative of1/(2+cos x) with help of tan x/2?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the antiderivative of the given function, hence, you need to find the function that differentiated yields `y = 1/(2+cos x).`

You need to use the reverse process of differentiation, hence, you need to use integration such that:

`int 1/(2+cos x) dx`

You need to use the function `tan (x/2),`  hence, you need to write the cosine function in terms of `tan(x/2)`  such that:

`cos x = (1- tan^2(x/2))/(1+ tan^2(x/2))`

You need to substitute `t `  for `tan (x/2)`  such that:

`tan (x/2) = t =gt dt = (t^2 + 1)/2 dx =gt dx = 2dt/(t^2 + 1)`

You need to write cos x in terms of t substituting t for tan(x/2) such that:

`cos x = (1-t^2)/(1+t^2)`

`int 1/(2+cos x) dx = int 1/(2 + (1-t^2)/(1+t^2)) *2dt/(t^2 + 1)`

`int 2dt/(2 + 2t^2 + 1 - t^2)= int 2dt/(3 + t^2)`

`int 2dt/(3 + t^2) = 2/sqrt3 tan^(-1) t/sqrt3 + c`

`int (dx)/(2 + cos x) = 2/sqrt3 tan^(-1) (tan(x/2))/sqrt3 + c`

Hence, evaluating the antiderivative of the given function using `tan(x/2)`  yields`int (dx)/(2 + cos x) = 2/sqrt3 tan^(-1) (tan(x/2))/sqrt3 + c.`


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