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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

\int e^{- 3 x} + 2 x d x

\int cos (t - x) d x

d er i v a t i v e f (x) = 7 x - 5

\frac{d}{d x} (\frac{1}{ln ( 2 )})

d er i v a t i v e f (x) = 1100 (x^{2} - 0.2 x)

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