What is the integral of (x^2+x+a)/(x^3-9x^2+27x-27) ?

The integral of (x^2+x+a)/(x^3-9x^2+27x-27) is ln|x-3|-7/(x-3)-6/((x-3)^2)-a/(2(x-3)^2)+C

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integral of (x^2+x+a)/(x^3-9x^2+27x-27)

\int \frac{x ^{2} + x + a}{x ^{3} - 9 x ^{2} + 27 x - 27} d x

ln ∣ x - 3 ∣ - \frac{7}{x - 3} - \frac{6}{( x - 3 ) ^{2}} - \frac{a}{2 ( x - 3 ) ^{2}} + C

Solution steps

\int \frac{x ^{2} + x + a}{x ^{3} - 9 x ^{2} + 27 x - 27} d x

Factor

\frac{x ^{2} + x + a}{x ^{3} - 9 x ^{2} + 27 x - 27}

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

Expand

\frac{x ^{2} + x + a}{( x - 3 ) ^{3}} : \frac{x ^{2}}{( x - 3 ) ^{3}} + \frac{x}{( x - 3 ) ^{3}} + \frac{a}{( x - 3 ) ^{3}}

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

\int \frac{x ^{2}}{( x - 3 ) ^{3}} d x = ln ∣ x - 3 ∣ - \frac{6}{x - 3} - \frac{9}{2 ( x - 3 ) ^{2}}

\int \frac{x}{( x - 3 ) ^{3}} d x = - \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}}

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

= ln ∣ x - 3 ∣ - \frac{6}{x - 3} - \frac{9}{2 ( x - 3 ) ^{2}} - \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}} - \frac{a}{2 ( x - 3 ) ^{2}}

Simplify

ln ∣ x - 3 ∣ - \frac{6}{x - 3} - \frac{9}{2 ( x - 3 ) ^{2}} - \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}} - \frac{a}{2 ( x - 3 ) ^{2}} : ln ∣ x - 3 ∣ - \frac{7}{x - 3} - \frac{6}{( x - 3 ) ^{2}} - \frac{a}{2 ( x - 3 ) ^{2}}

= ln ∣ x - 3 ∣ - \frac{7}{x - 3} - \frac{6}{( x - 3 ) ^{2}} - \frac{a}{2 ( x - 3 ) ^{2}}

Add a constant to the solution

= ln ∣ x - 3 ∣ - \frac{7}{x - 3} - \frac{6}{( x - 3 ) ^{2}} - \frac{a}{2 ( x - 3 ) ^{2}} + C

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What is the integral of (x^2+x+a)/(x^3-9x^2+27x-27) ?
The integral of (x^2+x+a)/(x^3-9x^2+27x-27) is ln|x-3|-7/(x-3)-6/((x-3)^2)-a/(2(x-3)^2)+C