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

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

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

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

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

Solution steps

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

Rewrite using trig identities

= \int (1 + cos (x)) csc (x) d x

Simplify

(1 + cos (x)) csc (x) : cot (x) + csc (x)

= \int cot (x) + csc (x) d 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 cot (x) d x + \int csc (x) d x

\int cot (x) d x = ln ∣ sin (x) ∣

\int csc (x) d x = ln tan (\frac{x}{2})

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

Add a constant to the solution

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

x \to 2 lim (\frac{3}{x + 1})

\int (2 x - 5)^{- \frac{1}{2}} d x

\frac{\partial}{\partial x} (y x^{3})

f (x) = \frac{x ^{3} + 7}{x}

d er i v a t i v e y = x^{e^{x}}

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