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

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

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

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

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

Solution steps

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

Long division

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

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

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

\int 1 d x = x

\int \frac{4 x - 2}{x ^{2} - x} d x = 2 ln x^{2} - x

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{7}{x ^{7}})

\int_{1}^{8} x^{- \frac{8}{3}} d x

y = x^{2} + 9, y = 10 + 2 x

x \to 3 lim (x^{2} - 9)

\int 3 cos (t) - 2 sin (t) d t

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