What is the integral of (cos^3(y))/(1-sin(y)) ?

The integral of (cos^3(y))/(1-sin(y)) is (sin^2(y))/2+sin(y)+C

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integral of (cos^3(y))/(1-sin(y))

\int \frac{cos ^{3} ( y )}{1 - sin ( y )} d y

\frac{sin ^{2} ( y )}{2} + sin (y) + C

Solution steps

\int \frac{cos ^{3} ( y )}{1 - sin ( y )} d y

Apply Trigonometric Substitution

= \int 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 u d u + \int 1 d u

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

\int 1 d u = u

= \frac{u ^{2}}{2} + u

Substitute back

u = sin (y)

= \frac{sin ^{2} ( y )}{2} + sin (y)

Add a constant to the solution

= \frac{sin ^{2} ( y )}{2} + sin (y) + C

What is the integral of (cos^3(y))/(1-sin(y)) ?
The integral of (cos^3(y))/(1-sin(y)) is (sin^2(y))/2+sin(y)+C