What is the integral of (1+sin(x))/(3+cos(x)) ?

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

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integral of (1+sin(x))/(3+cos(x))

\int \frac{1 + sin ( x )}{3 + cos ( x )} d x

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

Solution steps

\int \frac{1 + sin ( x )}{3 + cos ( x )} d x

Expand

\frac{1 + sin ( x )}{3 + cos ( x )} : \frac{1}{3 + cos ( x )} + \frac{sin ( x )}{3 + 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}{3 + cos ( x )} d x + \int \frac{sin ( x )}{3 + cos ( x )} d x

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

\int \frac{sin ( x )}{3 + cos ( x )} d x = - ln ∣ 3 + cos (x) ∣

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

Add a constant to the solution

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

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