Question

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

Accepted Answer

y= cosx /(sin x)^3
Let t= sinx
==> dt = cosx dx
=> y=  (dt / t^3)
intg y = intg (dt/t^3)
          = intg (t^-3) dt
           = -t^-2/2  + c
           = -1/2t^2 + C
Now substitute with t= sinx
==> intg y = -1/2(sin x)^2 + C

Answer

To find Integral {cosx/(sinx)^3} dx
Let us have a transformation  sinx = t. Then cosx dx = dt.
So Integral { cosxdx/ (sinx)^3} = Integral  dt/t^3
 = t^(-3+1)/(-3+1) +C
= t^-2/((-2) + C
= (sinx)^(-2) / 2 +
= 1/[2(sinx)^2] + const
So Int { cosxdx/(sinx)^3 } = 1/{2(sinx)^2} + constant

Answer

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.

Math

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

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