What is the integral of (1-sin(x))/(2+cos(x)) ?

The integral of (1-sin(x))/(2+cos(x)) is 2/(sqrt(3))arctan(1/(sqrt(3))tan(x/2))+ln|2+cos(x)|+C

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integral of (1-sin(x))/(2+cos(x))

\int \frac{1 - sin ( x )}{2 + cos ( x )} d x

\frac{2}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + ln ∣ 2 + cos (x) ∣ + C

Solution steps

\int \frac{1 - sin ( x )}{2 + cos ( x )} d x

Expand

\frac{1 - sin ( x )}{2 + cos ( x )} : \frac{1}{2 + cos ( x )} - \frac{sin ( x )}{2 + cos ( x )}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{2 + cos ( x )} d x - \int \frac{sin ( x )}{2 + cos ( x )} d x

\int \frac{1}{2 + cos ( x )} d x = \frac{2}{3} arctan (\frac{1}{3} tan (\frac{x}{2}))

\int \frac{sin ( x )}{2 + cos ( x )} d x = - ln ∣ 2 + cos (x) ∣

= \frac{2}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) - (- ln ∣ 2 + cos (x) ∣)

Simplify

= \frac{2}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + ln ∣ 2 + cos (x) ∣

Add a constant to the solution

= \frac{2}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + ln ∣ 2 + cos (x) ∣ + C

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What is the integral of (1-sin(x))/(2+cos(x)) ?
The integral of (1-sin(x))/(2+cos(x)) is 2/(sqrt(3))arctan(1/(sqrt(3))tan(x/2))+ln|2+cos(x)|+C