What is the integral of (sin(x))/((1-sin(x))^2) ?

The integral of (sin(x))/((1-sin(x))^2) is 2/3 sec^3(x)+2/3 tan^3(x)-sec(x)+C

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integral of (sin(x))/((1-sin(x))^2)

\int \frac{sin ( x )}{( 1 - sin ( x ) ) ^{2}} d x

\frac{2}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) - sec (x) + C

Solution steps

\int \frac{sin ( x )}{( 1 - sin ( x ) ) ^{2}} 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 )} ) ^{2}} d x

Expand

\frac{sin ( x )}{( \frac{c o s ^{2} ( x )}{1 + s i n ( x )} ) ^{2}} : \frac{sin ( x )}{cos ^{4} ( x )} + \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} + \frac{sin ^{3} ( x )}{cos ^{4} ( x )}

= \int \frac{sin ( x )}{cos ^{4} ( x )} + \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} + \frac{sin ^{3} ( x )}{cos ^{4} ( 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 ^{4} ( x )} d x + \int \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} d x + \int \frac{sin ^{3} ( x )}{cos ^{4} ( x )} d x

\int \frac{sin ( x )}{cos ^{4} ( x )} d x = \frac{1}{3} sec^{3} (x)

\int \frac{2 sin ^{2} ( x )}{cos ^{4} ( x )} d x = \frac{2}{3} tan^{3} (x)

\int \frac{sin ^{3} ( x )}{cos ^{4} ( x )} d x = \frac{1}{3} sec^{3} (x) - sec (x)

= \frac{1}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) + \frac{1}{3} sec^{3} (x) - sec (x)

Add similar elements:

\frac{1}{3} sec^{3} (x) + \frac{1}{3} sec^{3} (x) = \frac{2}{3} sec^{3} (x)

= \frac{2}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) - sec (x)

Add a constant to the solution

= \frac{2}{3} sec^{3} (x) + \frac{2}{3} tan^{3} (x) - sec (x) + C

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What is the integral of (sin(x))/((1-sin(x))^2) ?
The integral of (sin(x))/((1-sin(x))^2) is 2/3 sec^3(x)+2/3 tan^3(x)-sec(x)+C