What is the integral of (cos(x))/(sin(x)-1) ?

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

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integral of (cos(x))/(sin(x)-1)

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

ln ∣ sin (x) - 1 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = sin (x) - 1

= ln ∣ sin (x) - 1 ∣

Add a constant to the solution

= ln ∣ sin (x) - 1 ∣ + C

\int \frac{1}{( sin ( x ) )} d x

\int \frac{9}{( 1 - 3 x ) ^{4}} d x

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

\int 4 arctan (4 x) d x

y

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