Calculate the indefinite integral of y = cos x / (sin x)^3.

To calculate the indefinite integral, we'll use the
substitution method.

We'll note y =
f(x)

We'll calculate Integral of f(x) = y = cos x / (sin
x)^3.

We notice that if we'll differentiate sin x, we'll
get cos x.

So, we'll note sin x =
t

(sin x)'dx = dt

(cos x)dx =
dt

We'll re-write the integral in the variable
t:

Int  (cos x)dx / (sin x)^3 = Int dt /
t^3

Int dt / t^3 = Int
[t^(-3)]dt

Int [t^(-3)]dt = t^(-3+1) / (-3+1) +
C

Int [t^(-3)]dt = t^(-2)/-2 +
C

Int [t^(-3)]dt = -1 / 2t^2 +
C

But sin x =
t.

Int  (cos x)dx / (sin x)^3 = -1 / 2(sin
x)^2 + C, where C is a family of constants.