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Find the integral `int (2*cos x)/(sin x + cos x) dx`

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shayaan | Student, Undergraduate | (Level 1) Honors

Posted December 28, 2011 at 2:57 AM via web

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Find the integral `int (2*cos x)/(sin x + cos x) dx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted December 28, 2011 at 3:07 AM (Answer #1)

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The integral `int (2*cos x)/(sin x + cos x) dx` has to be determined.

2*cos x = sin x + cos x + cos x - sin x

Substituting this in the integral gives

`int (sin x+cos x+cos x - sin x)/(sin x + cos x) dx`

=> `int (sin x + cos x)/(sin x + cos x) - (sin x - cos x)/(sin x + cos x) dx`

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

=> `x - int (sin x - cos x)/(sin x + cos x) dx`

Let sin x + cos x = y

`dy/dx = cos x - sin x`

=> `dy = (cos x - sin x) dx`

substituting in the integral

=> `x + int (1/y) dy`

=> `x + ln y`

substitute y = cos x + sin x

=> `x + ln(cos x + sin x)`

The integral` ` `int (2*cos x)/(sin x + cos x) dx = x + ln(cos x + sin x)`

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