What is the integral of (x^2-x-20)/(x-5) ?

The integral of (x^2-x-20)/(x-5) is 9x+1/2 (x-5)^2-45+C

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integral of (x^2-x-20)/(x-5)

\int \frac{x ^{2} - x - 20}{x - 5} d x

9 x + \frac{1}{2} (x - 5)^{2} - 45 + C

Solution steps

\int \frac{x ^{2} - x - 20}{x - 5} d x

Expand

\frac{x ^{2} - x - 20}{x - 5} : \frac{x ^{2}}{x - 5} - \frac{x}{x - 5} - \frac{20}{x - 5}

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 - 5} d x - \int \frac{x}{x - 5} d x - \int \frac{20}{x - 5} d x

\int \frac{x ^{2}}{x - 5} d x = \frac{( x - 5 ) ^{2}}{2} + 10 (x - 5) + 25 ln ∣ x - 5 ∣

\int \frac{x}{x - 5} d x = x - 5 + 5 ln ∣ x - 5 ∣

\int \frac{20}{x - 5} d x = 20 ln ∣ x - 5 ∣

= \frac{( x - 5 ) ^{2}}{2} + 10 (x - 5) + 25 ln ∣ x - 5 ∣ - (x - 5 + 5 ln ∣ x - 5 ∣) - 20 ln ∣ x - 5 ∣

Simplify

\frac{( x - 5 ) ^{2}}{2} + 10 (x - 5) + 25 ln ∣ x - 5 ∣ - (x - 5 + 5 ln ∣ x - 5 ∣) - 20 ln ∣ x - 5 ∣ : 9 x + \frac{1}{2} (x - 5)^{2} - 45

= 9 x + \frac{1}{2} (x - 5)^{2} - 45

Add a constant to the solution

= 9 x + \frac{1}{2} (x - 5)^{2} - 45 + C

What is the integral of (x^2-x-20)/(x-5) ?
The integral of (x^2-x-20)/(x-5) is 9x+1/2 (x-5)^2-45+C