What is the integral of (2x^2-5x-7)/(x-3) ?

The integral of (2x^2-5x-7)/(x-3) is x^2+x-4ln|x-3|-12+C

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integral of (2x^2-5x-7)/(x-3)

\int \frac{2 x ^{2} - 5 x - 7}{x - 3} d x

x^{2} + x - 4 ln ∣ x - 3 ∣ - 12 + C

Solution steps

\int \frac{2 x ^{2} - 5 x - 7}{x - 3} d x

Expand

\frac{2 x ^{2} - 5 x - 7}{x - 3} : \frac{2 x ^{2}}{x - 3} - \frac{5 x}{x - 3} - \frac{7}{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{2 x ^{2}}{x - 3} d x - \int \frac{5 x}{x - 3} d x - \int \frac{7}{x - 3} d x

\int \frac{2 x ^{2}}{x - 3} d x = x^{2} + 6 x + 18 ln ∣ x - 3 ∣ - 27

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

\int \frac{7}{x - 3} d x = 7 ln ∣ x - 3 ∣

= x^{2} + 6 x + 18 ln ∣ x - 3 ∣ - 27 - 5 (x - 3 + 3 ln ∣ x - 3 ∣) - 7 ln ∣ x - 3 ∣

Simplify

x^{2} + 6 x + 18 ln ∣ x - 3 ∣ - 27 - 5 (x - 3 + 3 ln ∣ x - 3 ∣) - 7 ln ∣ x - 3 ∣ : x^{2} + x - 4 ln ∣ x - 3 ∣ - 12

= x^{2} + x - 4 ln ∣ x - 3 ∣ - 12

Add a constant to the solution

= x^{2} + x - 4 ln ∣ x - 3 ∣ - 12 + C

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What is the integral of (2x^2-5x-7)/(x-3) ?
The integral of (2x^2-5x-7)/(x-3) is x^2+x-4ln|x-3|-12+C