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

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

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

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

x - \frac{1}{2} ln ∣ sin (2 x) ∣ + C

Solution steps

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

Expand

\frac{sin ( 2 x ) - cos ( 2 x )}{sin ( 2 x )} : 1 - \frac{cos ( 2 x )}{sin ( 2 x )}

= \int 1 - \frac{cos ( 2 x )}{sin ( 2 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 1 d x - \int \frac{cos ( 2 x )}{sin ( 2 x )} d x

\int 1 d x = x

\int \frac{cos ( 2 x )}{sin ( 2 x )} d x = \frac{1}{2} ln ∣ sin (2 x) ∣

= x - \frac{1}{2} ln ∣ sin (2 x) ∣

Add a constant to the solution

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

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