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

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

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

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

x - ln ∣ sin (x) ∣ + C

Solution steps

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

Expand

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

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

\int 1 d x = x

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

= x - ln ∣ sin (x) ∣

Add a constant to the solution

= x - ln ∣ sin (x) ∣ + C

\int ln (x^{2} + 2 x + 2) d x

\int \frac{x ^{4} - 3 x ^{2} - 3 x - 2}{x ^{3} - x ^{2} - 2 x} d x

\int \frac{3 x + 4}{3 x - 2} d x

\int \frac{5}{x ( ln ( x ) ) ^{2}} d x

\int x (7 - x)^{11} d x

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