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

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

in t e g r a l y

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