What is the integral of (x^2-3x+11)/(x-2) ?

The integral of (x^2-3x+11)/(x-2) is ((x-2)^2)/2+x-2+9ln|x-2|+C

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integral of (x^2-3x+11)/(x-2)

\int \frac{x ^{2} - 3 x + 11}{x - 2} d x

\frac{( x - 2 ) ^{2}}{2} + x - 2 + 9 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x ^{2} - 3 x + 11}{x - 2} d x

Apply u-substitution

= \int \frac{u ^{2} + u + 9}{u} d u

Expand

\frac{u ^{2} + u + 9}{u} : u + 1 + \frac{9}{u}

= \int u + 1 + \frac{9}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u d u + \int 1 d u + \int \frac{9}{u} d u

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

\int 1 d u = u

\int \frac{9}{u} d u = 9 ln ∣ u ∣

= \frac{u ^{2}}{2} + u + 9 ln ∣ u ∣

Substitute back

u = x - 2

= \frac{( x - 2 ) ^{2}}{2} + x - 2 + 9 ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{( x - 2 ) ^{2}}{2} + x - 2 + 9 ln ∣ x - 2 ∣ + C

What is the integral of (x^2-3x+11)/(x-2) ?
The integral of (x^2-3x+11)/(x-2) is ((x-2)^2)/2+x-2+9ln|x-2|+C