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

The integral of ((cos(3t)+3))/(sin(3t)) is 1/3 ln|sin(3t)|+ln|tan((3t)/2)|+C

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integral of ((cos(3t)+3))/(sin(3t))

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

\frac{1}{3} ln ∣ sin (3 t) ∣ + ln tan (\frac{3 t}{2}) + C

Solution steps

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

Expand

\frac{cos ( 3 t ) + 3}{sin ( 3 t )} : \frac{cos ( 3 t )}{sin ( 3 t )} + \frac{3}{sin ( 3 t )}

= \int \frac{cos ( 3 t )}{sin ( 3 t )} + \frac{3}{sin ( 3 t )} d t

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

\int \frac{cos ( 3 t )}{sin ( 3 t )} d t = \frac{1}{3} ln ∣ sin (3 t) ∣

\int \frac{3}{sin ( 3 t )} d t = ln tan (\frac{3 t}{2})

= \frac{1}{3} ln ∣ sin (3 t) ∣ + ln tan (\frac{3 t}{2})

Add a constant to the solution

= \frac{1}{3} ln ∣ sin (3 t) ∣ + ln tan (\frac{3 t}{2}) + C

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