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

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

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integral of 2/(x^3-x^2)

\int \frac{2}{x ^{3} - x ^{2}} d x

2 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣) + C

Solution steps

\int \frac{2}{x ^{3} - x ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{x ^{3} - x ^{2}} d x

Take the partial fraction of

\frac{1}{x ^{3} - x ^{2}} : - \frac{1}{x} - \frac{1}{x ^{2}} + \frac{1}{x - 1}

= 2 \cdot \int - \frac{1}{x} - \frac{1}{x ^{2}} + \frac{1}{x - 1} d x

Apply the Sum Rule:

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= 2 (- ln ∣ x ∣ - (- \frac{1}{x}) + ln ∣ x - 1 ∣)

Simplify

= 2 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣)

Add a constant to the solution

= 2 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣) + C

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