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

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

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

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

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

Solution steps

\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}

= \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

= - \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 ∣

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

Simplify

= - ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣

Add a constant to the solution

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

\int e^{c o s (29 t)} sin (29 t) d t

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

f (x) = - x + \frac{6}{x} + 1

\frac{s - 1}{s ^{2} - 2 s + 5}

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