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

The integral of (x^3-4x+1)/(x^3-2x^2) is x+7/4 ln|x|+1/4 ln|x-2|+1/(2x)-2+C

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

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

x + \frac{7}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x - 2 ∣ + \frac{1}{2 x} - 2 + C

Solution steps

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

Expand

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

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

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

\int \frac{4 x}{x ^{3} - 2 x ^{2}} d x = 4 (- \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣)

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

= x - 2 + 2 ln ∣ x - 2 ∣ - 4 (- \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣) - \frac{1}{4} ln ∣ x ∣ + \frac{1}{2 x} + \frac{1}{4} ln ∣ x - 2 ∣

Simplify

x - 2 + 2 ln ∣ x - 2 ∣ - 4 (- \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣) - \frac{1}{4} ln ∣ x ∣ + \frac{1}{2 x} + \frac{1}{4} ln ∣ x - 2 ∣ : x + \frac{7}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x - 2 ∣ + \frac{1}{2 x} - 2

= x + \frac{7}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x - 2 ∣ + \frac{1}{2 x} - 2

Add a constant to the solution

= x + \frac{7}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x - 2 ∣ + \frac{1}{2 x} - 2 + C

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What is the integral of (x^3-4x+1)/(x^3-2x^2) ?
The integral of (x^3-4x+1)/(x^3-2x^2) is x+7/4 ln|x|+1/4 ln|x-2|+1/(2x)-2+C