Integrate the given function defined over a proper domain w.r.t. x : $\frac{x^{3} - 6 x^{2} + 10 x - 2}{x^{2} - 5 x + 6}$

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Solution

$\int \frac{x^{3} - 6 x^{2} + 10 x - 2}{x^{2} - 5 x + 6} d x$
$= \int (x - 1) d x + \int \frac{- x + 4}{x^{2} - 5 x + 6} d x$
$= \frac{x^{2}}{2} - x - \int \frac{x - 4}{x^{2} - 5 x + 6} d x$
$= \frac{x^{2}}{2} - x - \int \frac{x - 3 - 1}{x^{2} - 5 x + 6} d x$
$= \frac{x^{2}}{2} - x - \int \frac{x - 3}{x^{2} - 5 x + 6} d x + \int \frac{d x}{x^{2} - 5 x + 6}$
$= \frac{x^{2}}{2} - x - \int \frac{x - 3}{(x - 3) (x - 2)} d x + \int \frac{[(x - 3) - (x - 2)] d x}{(x - 3) (x - 2)}$
$= \frac{x^{2}}{2} - x - \int \frac{d x}{(x - 2)} + \int \frac{[(x - 3) - (x - 2)] d x}{(x - 3) (x - 2)}$
$= \frac{x^{2}}{2} - x - log | x - 2 | + \int \frac{(x - 3) d x}{(x - 3) (x - 2)} - \int \frac{(x - 2) d x}{(x - 3) (x - 2)}$
$= \frac{x^{2}}{2} - x - log | x - 2 | + \int \frac{d x}{(x - 2)} - \int \frac{d x}{(x - 3)}$
$= \frac{x^{2}}{2} - x - log | x - 2 | + log | x - 2 | - log | x - 3 | + c$
$= \frac{x^{2}}{2} - x - log | x - 3 | + c$

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Similar questions

\frac{x^{3} - 6 x^{2} + 10 x - 2}{x^{2} - 5 x + 6}

\frac{x^{3} - 6 x^{2} + 10 x - 2}{x^{2} - 5 x + 6} = f (x) + \frac{A}{x - 2} + \frac{B}{x - 3}

f (x) =

A, B

f (x) = \frac{1}{\sqrt{5 x - x^{2} - 6}}

\frac{x^{3} - 27}{x^{2} + 10 x + 24} \times \frac{x^{2} + 4 x - 12}{x^{2} - 5 x + 6} .

f (x) = \frac{^{3} \sqrt{x^{2} - 5 x + 6}}{x^{2} - x - 6}

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