Find the integral integrate of ((cos(x))^2)/(1+ sin(x))dx

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use the fundamental formula of trigonometry such that:

`cos^2 x = 1  - sin^2 x`

You may convert the difference of squares into a product such that:

`1 - sin^2 x = (1 - sin x)(1 + sin x)`

You need to substitute `(1 - sin x)(1 + sin x)`  for `cos^2 x`  such that:

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

Reducing by `1 + sin x`  yields:

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

Integrating both sides yields:

`int (cos^2 x)/(1 + sin x) dx= int (1 - sin x) dx`

Using the linearity yields:

`int (cos^2 x)/(1 + sin x) dx = intdx - int sin x dx`

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

`int (cos^2 x)/(1 + sin x) dx = x + cos x + c`

Hence, evaluating the given integral under the given conditions yields `int (cos^2 x)/(1 + sin x) dx = x + cos x + c.`

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