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

The integral of (sin(x))/(1-sin(x)) is sec(x)+tan(x)-x+C

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

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

sec (x) + tan (x) - x + C

Solution steps

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

Multiply by the conjugate of

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

= \int \frac{sin ( x )}{\frac{c o s ^{2} ( x )}{1 + s i n ( x )}} d x

Expand

\frac{sin ( x )}{\frac{c o s ^{2} ( x )}{1 + s i n ( x )}} : \frac{sin ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

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

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

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

= sec (x) + tan (x) - x

Add a constant to the solution

= sec (x) + tan (x) - x + C

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What is the integral of (sin(x))/(1-sin(x)) ?
The integral of (sin(x))/(1-sin(x)) is sec(x)+tan(x)-x+C