What is the integral of (2x^4-2x^3+x+1)/(x^3-x^2) ?

The integral of (2x^4-2x^3+x+1)/(x^3-x^2) is x^2-2ln|x|+1/x+2ln|x-1|+C

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integral of (2x^4-2x^3+x+1)/(x^3-x^2)

\int \frac{2 x ^{4} - 2 x ^{3} + x + 1}{x ^{3} - x ^{2}} d x

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

Solution steps

\int \frac{2 x ^{4} - 2 x ^{3} + x + 1}{x ^{3} - x ^{2}} d x

Long division

\frac{2 x ^{4} - 2 x ^{3} + x + 1}{x ^{3} - x ^{2}} : 2 x + \frac{x + 1}{x ^{3} - x ^{2}}

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

\int 2 x d x = x^{2}

\int \frac{x + 1}{x ^{3} - x ^{2}} d x = - 2 ln ∣ x ∣ + \frac{1}{x} + 2 ln ∣ x - 1 ∣

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

Add a constant to the solution

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

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What is the integral of (2x^4-2x^3+x+1)/(x^3-x^2) ?
The integral of (2x^4-2x^3+x+1)/(x^3-x^2) is x^2-2ln|x|+1/x+2ln|x-1|+C