What is the integral of (3x^2-7x-24)/(x-2) ?

The integral of (3x^2-7x-24)/(x-2) is 5x+3/2 (x-2)^2-26ln|x-2|-10+C

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integral of (3x^2-7x-24)/(x-2)

\int \frac{3 x ^{2} - 7 x - 24}{x - 2} d x

5 x + \frac{3}{2} (x - 2)^{2} - 26 ln ∣ x - 2 ∣ - 10 + C

Solution steps

\int \frac{3 x ^{2} - 7 x - 24}{x - 2} d x

Expand

\frac{3 x ^{2} - 7 x - 24}{x - 2} : \frac{3 x ^{2}}{x - 2} - \frac{7 x}{x - 2} - \frac{24}{x - 2}

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{3 x ^{2}}{x - 2} d x - \int \frac{7 x}{x - 2} d x - \int \frac{24}{x - 2} d x

\int \frac{3 x ^{2}}{x - 2} d x = 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣)

\int \frac{7 x}{x - 2} d x = 7 (x - 2 + 2 ln ∣ x - 2 ∣)

\int \frac{24}{x - 2} d x = 24 ln ∣ x - 2 ∣

= 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) - 7 (x - 2 + 2 ln ∣ x - 2 ∣) - 24 ln ∣ x - 2 ∣

Simplify

3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) - 7 (x - 2 + 2 ln ∣ x - 2 ∣) - 24 ln ∣ x - 2 ∣ : 5 x + \frac{3}{2} (x - 2)^{2} - 26 ln ∣ x - 2 ∣ - 10

= 5 x + \frac{3}{2} (x - 2)^{2} - 26 ln ∣ x - 2 ∣ - 10

Add a constant to the solution

= 5 x + \frac{3}{2} (x - 2)^{2} - 26 ln ∣ x - 2 ∣ - 10 + C

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What is the integral of (3x^2-7x-24)/(x-2) ?
The integral of (3x^2-7x-24)/(x-2) is 5x+3/2 (x-2)^2-26ln|x-2|-10+C