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

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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

\frac{\partial}{\partial x} (e^{x} + y + y^{4} + 6)

y^{^{''}} + \frac{3}{x} y^{^{'}} - \frac{8}{x ^{2}} y = 0

\int_{- 5}^{5} \frac{1}{x} d x

y^{^{''}} + 10 y^{^{'}} + 25 y = t^{- 2} e^{- 5 t}

\frac{\partial}{\partial x} (e^{2 x} cos (3 y))

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