What is the integral of (x^2-5x-24)/(x+3) ?

The integral of (x^2-5x-24)/(x+3) is -11x+1/2 (x+3)^2-33+C

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integral of (x^2-5x-24)/(x+3)

\int \frac{x ^{2} - 5 x - 24}{x + 3} d x

- 11 x + \frac{1}{2} (x + 3)^{2} - 33 + C

Solution steps

\int \frac{x ^{2} - 5 x - 24}{x + 3} d x

Expand

\frac{x ^{2} - 5 x - 24}{x + 3} : \frac{x ^{2}}{x + 3} - \frac{5 x}{x + 3} - \frac{24}{x + 3}

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} d x - \int \frac{5 x}{x + 3} d x - \int \frac{24}{x + 3} d x

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

\int \frac{5 x}{x + 3} d x = 5 (x + 3 - 3 ln ∣ x + 3 ∣)

\int \frac{24}{x + 3} d x = 24 ln ∣ x + 3 ∣

= \frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣ - 5 (x + 3 - 3 ln ∣ x + 3 ∣) - 24 ln ∣ x + 3 ∣

Simplify

\frac{( x + 3 ) ^{2}}{2} - 6 (x + 3) + 9 ln ∣ x + 3 ∣ - 5 (x + 3 - 3 ln ∣ x + 3 ∣) - 24 ln ∣ x + 3 ∣ : - 11 x + \frac{1}{2} (x + 3)^{2} - 33

= - 11 x + \frac{1}{2} (x + 3)^{2} - 33

Add a constant to the solution

= - 11 x + \frac{1}{2} (x + 3)^{2} - 33 + C

What is the integral of (x^2-5x-24)/(x+3) ?
The integral of (x^2-5x-24)/(x+3) is -11x+1/2 (x+3)^2-33+C