What is the integral of 1/(y^2-y^3) ?

The integral of 1/(y^2-y^3) is ln|y|-1/y-ln|y-1|+C

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integral of 1/(y^2-y^3)

\int \frac{1}{y ^{2} - y ^{3}} d y

ln ∣ y ∣ - \frac{1}{y} - ln ∣ y - 1 ∣ + C

Solution steps

\int \frac{1}{y ^{2} - y ^{3}} d y

Take the partial fraction of

\frac{1}{y ^{2} - y ^{3}} : - \frac{- y - 1}{y ^{2}} - \frac{1}{y - 1}

= \int - \frac{- y - 1}{y ^{2}} - \frac{1}{y - 1} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{- y - 1}{y ^{2}} d y - \int \frac{1}{y - 1} d y

\int \frac{- y - 1}{y ^{2}} d y = - ln ∣ y ∣ + \frac{1}{y}

\int \frac{1}{y - 1} d y = ln ∣ y - 1 ∣

= - (- ln ∣ y ∣ + \frac{1}{y}) - ln ∣ y - 1 ∣

Simplify

= ln ∣ y ∣ - \frac{1}{y} - ln ∣ y - 1 ∣

Add a constant to the solution

= ln ∣ y ∣ - \frac{1}{y} - ln ∣ y - 1 ∣ + C

\frac{d}{d x} (2 x + \frac{100}{x})

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

\frac{x}{x + 3}

\int_{0}^{2} 5^{- x} d x

x y^{^{'}} + 15 y = 0

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