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

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

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

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

- \frac{cos ^{2} ( x )}{2} - cos (x) + C

Solution steps

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

Apply u-substitution

: \int - u - 1 d u

= \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 = cos (x)

= - \frac{cos ^{2} ( x )}{2} - cos (x)

Add a constant to the solution

= - \frac{cos ^{2} ( x )}{2} - cos (x) + C

\int x - \frac{1}{x} d x

\int (e^{2 x} cos (e^{2 x})) d x

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

x \frac{d y}{d x} - (1 + x) y = x y^{2}

f (t) = \frac{t ^{2} - 4}{t - 2}

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