What is the integral of ((2x^2+9x-6)/(x-5)) ?

The integral of ((2x^2+9x-6)/(x-5)) is x^2+19x+89ln|x-5|-120+C

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integral of ((2x^2+9x-6)/(x-5))

\int (\frac{2 x ^{2} + 9 x - 6}{x - 5}) d x

x^{2} + 19 x + 89 ln ∣ x - 5 ∣ - 120 + C

Solution steps

\int \frac{2 x ^{2} + 9 x - 6}{x - 5} d x

Apply u-substitution

: \int \frac{2 u ^{2} + 29 u + 89}{u} d u

= \int \frac{2 u ^{2} + 29 u + 89}{u} d u

Expand

\frac{2 u ^{2} + 29 u + 89}{u} : 2 u + 29 + \frac{89}{u}

= \int 2 u + 29 + \frac{89}{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 2 u d u + \int 29 d u + \int \frac{89}{u} d u

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

\int 29 d u = 29 u

\int \frac{89}{u} d u = 89 ln ∣ u ∣

= u^{2} + 29 u + 89 ln ∣ u ∣

Substitute back

u = x - 5

= (x - 5)^{2} + 29 (x - 5) + 89 ln ∣ x - 5 ∣

Simplify

(x - 5)^{2} + 29 (x - 5) + 89 ln ∣ x - 5 ∣ : x^{2} + 19 x + 89 ln ∣ x - 5 ∣ - 120

= x^{2} + 19 x + 89 ln ∣ x - 5 ∣ - 120

Add a constant to the solution

= x^{2} + 19 x + 89 ln ∣ x - 5 ∣ - 120 + C

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What is the integral of ((2x^2+9x-6)/(x-5)) ?
The integral of ((2x^2+9x-6)/(x-5)) is x^2+19x+89ln|x-5|-120+C