What is the integral of (cos(x)+sin(2x))/(sin(x)) ?

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

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integral of (cos(x)+sin(2x))/(sin(x))

\int \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

ln ∣ sin (x) ∣ + 2 sin (x) + C

Solution steps

\int \frac{cos ( x ) + sin ( 2 x )}{sin ( x )} d x

Expand

\frac{cos ( x ) + sin ( 2 x )}{sin ( x )} : \frac{cos ( x )}{sin ( x )} + \frac{sin ( 2 x )}{sin ( x )}

= \int \frac{cos ( x )}{sin ( x )} + \frac{sin ( 2 x )}{sin ( 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 ( x )}{sin ( x )} d x + \int \frac{sin ( 2 x )}{sin ( x )} d x

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

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

= ln ∣ sin (x) ∣ + 2 sin (x)

Add a constant to the solution

= ln ∣ sin (x) ∣ + 2 sin (x) + C

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