Calculate F(x) if f'(x)=cosx/(sinx)^3

To determine the function F(x) = f(x), we'll have to
integrate cosx/(sinx)^3.

Int cosxdx/(sinx)^3 = F(x) +
C

To determine the integral of f'(x), we'll change the
variable x.

We'll note sin x =
t

We'll differentiate both
sides:

cos xdx = dt

We'll
substitute sin x by t and cosxdx by dt and we'll get:

Int
cosxdx/(sinx)^3 = Int dt/t^3

We'll use the property of the
negative exponent:

1/t^3 =
t^-3

Int dt/t^3  = Int
t^-3dt

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

Int t^-3dt = t^-2/-2 + C

Int
t^-3dt = -1/2t^2 + C

But t = sin
x

Int cosxdx/(sinx)^3 = 1/2(sin x)^2 +
C

So, F(x) =  1/2(sin x)^2 +
C.