What is the integral of (x^4-6)/(x^3-2x^2) ?

The integral of (x^4-6)/(x^3-2x^2) is (x^2)/2+2x+3/2 ln|x|-3/x+5/2 ln|x-2|+C

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integral of (x^4-6)/(x^3-2x^2)

\int \frac{x ^{4} - 6}{x ^{3} - 2 x ^{2}} d x

\frac{x ^{2}}{2} + 2 x + \frac{3}{2} ln ∣ x ∣ - \frac{3}{x} + \frac{5}{2} ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x ^{4} - 6}{x ^{3} - 2 x ^{2}} d x

Factor

\frac{x ^{4} - 6}{x ^{3} - 2 x ^{2}}

Take the partial fraction of

= \int x + 2 + \frac{3}{2 x} + \frac{3}{x ^{2}} + \frac{5}{2 ( 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 x d x + \int 2 d x + \int \frac{3}{2 x} d x + \int \frac{3}{x ^{2}} d x + \int \frac{5}{2 ( x - 2 )} d x

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

\int 2 d x = 2 x

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

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

\int \frac{5}{2 ( x - 2 )} d x = \frac{5}{2} ln ∣ x - 2 ∣

= \frac{x ^{2}}{2} + 2 x + \frac{3}{2} ln ∣ x ∣ - \frac{3}{x} + \frac{5}{2} ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 2 x + \frac{3}{2} ln ∣ x ∣ - \frac{3}{x} + \frac{5}{2} ln ∣ x - 2 ∣ + C

What is the integral of (x^4-6)/(x^3-2x^2) ?
The integral of (x^4-6)/(x^3-2x^2) is (x^2)/2+2x+3/2 ln|x|-3/x+5/2 ln|x-2|+C