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

The integral of (2-sin(x))/(2+cos(x)) is 4/(sqrt(3))arctan(1/(sqrt(3))tan(x/2))+ln|2+cos(x)|+C

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

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

\frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + ln ∣ 2 + cos (x) ∣ + C

Solution steps

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

Expand

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

\int \frac{2}{2 + cos ( x )} d x = \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2}))

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

= \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) - (- ln ∣ 2 + cos (x) ∣)

Simplify

= \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + ln ∣ 2 + cos (x) ∣

Add a constant to the solution

= \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + ln ∣ 2 + cos (x) ∣ + C

cos (3 x)

\int 60 d x

\int x^{4} x + 1 d x

f (x) = (3 x + 1)^{2}

\int_{0}^{2} \int_{3 y}^{6} 5 e^{x^{2}} d x d y

What is the integral of (2-sin(x))/(2+cos(x)) ?
The integral of (2-sin(x))/(2+cos(x)) is 4/(sqrt(3))arctan(1/(sqrt(3))tan(x/2))+ln|2+cos(x)|+C