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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Simplify

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

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

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 (a) d a + \int csc (a) d a

\int cot (a) d a = ln ∣ sin (a) ∣

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

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

Add a constant to the solution

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

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