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

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

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

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

- \frac{2}{tan ( \frac{x}{2} ) + 1} + ln ∣ 1 + sin (x) ∣ + C

Solution steps

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

Expand

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

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

\int \frac{cos ( x )}{1 + sin ( x )} d x = ln ∣ 1 + sin (x) ∣

= - \frac{2}{tan ( \frac{x}{2} ) + 1} + ln ∣ 1 + sin (x) ∣

Add a constant to the solution

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

\frac{\partial}{\partial x} (5 x^{4} + 2 x y + 1)

e^{2 t} cos (3 t)

\int + \frac{x ^{4} + 12 x ^{2} + 1}{x ^{5} + 20 x ^{3} + 5 x} d x

\int 2 e^{x} - 8 x d x

y = \frac{5}{1 + x ^{2}}, (4, 0.29412)

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