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

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

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

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

x + ln ∣ x ∣ + C

Solution steps

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

Simplify

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

= \int \frac{x + 1}{x} d x

Expand

\frac{x + 1}{x} : 1 + \frac{1}{x}

= \int 1 + \frac{1}{x} 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 1 d x + \int \frac{1}{x} d x

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

= x + ln ∣ x ∣

Add a constant to the solution

= x + ln ∣ x ∣ + C

\int_{5}^{10} \frac{8}{5 - 2 x} d x

y = \frac{1}{3} (x^{2} + 2)^{\frac{3}{2}}

sec (9 x)

\frac{d}{d x} (4.5 x + 2.4 x^{2} + 1.1 x^{3})

\frac{d}{d x} (ln (x y))

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