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

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

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

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

- ln ∣ sin (x) ∣ + ln ∣ sin (x) - 1 ∣ + C

Solution steps

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

Apply u-substitution

= \int \frac{1}{u ^{2} - u} d u

Take the partial fraction of

\frac{1}{u ^{2} - u} : - \frac{1}{u} + \frac{1}{u - 1}

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

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{1}{u} d u + \int \frac{1}{u - 1} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

= - ln ∣ u ∣ + ln ∣ u - 1 ∣

Substitute back

u = sin (x)

= - ln ∣ sin (x) ∣ + ln ∣ sin (x) - 1 ∣

Add a constant to the solution

= - ln ∣ sin (x) ∣ + ln ∣ sin (x) - 1 ∣ + C

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