What is the integral of (3x^3-9x^2+9x+3)/(3x-3) ?

The integral of (3x^3-9x^2+9x+3)/(3x-3) is (x^3)/3-x^2+x+2ln|x-1|+C

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integral of (3x^3-9x^2+9x+3)/(3x-3)

\int \frac{3 x ^{3} - 9 x ^{2} + 9 x + 3}{3 x - 3} d x

\frac{x ^{3}}{3} - x^{2} + x + 2 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{3 x ^{3} - 9 x ^{2} + 9 x + 3}{3 x - 3} d x

Long division

\frac{3 x ^{3} - 9 x ^{2} + 9 x + 3}{3 x - 3} : x^{2} - 2 x + 1 + \frac{2}{x - 1}

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

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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

\int x ln (9 x) d x

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\int \frac{( x ^{4} )}{1 - x ^{10}} d x

\int a x - x^{2} d x

\int \frac{cos ^{3} ( x )}{3} d x

What is the integral of (3x^3-9x^2+9x+3)/(3x-3) ?
The integral of (3x^3-9x^2+9x+3)/(3x-3) is (x^3)/3-x^2+x+2ln|x-1|+C