What is the integral of (2x^2-16x+30)/(x-5) ?

The integral of (2x^2-16x+30)/(x-5) is x^2-6x+5+C

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integral of (2x^2-16x+30)/(x-5)

\int \frac{2 x ^{2} - 16 x + 30}{x - 5} d x

x^{2} - 6 x + 5 + C

Solution steps

\int \frac{2 x ^{2} - 16 x + 30}{x - 5} d x

Expand

\frac{2 x ^{2} - 16 x + 30}{x - 5} : \frac{2 x ^{2}}{x - 5} - \frac{16 x}{x - 5} + \frac{30}{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{2 x ^{2}}{x - 5} d x - \int \frac{16 x}{x - 5} d x + \int \frac{30}{x - 5} d x

\int \frac{2 x ^{2}}{x - 5} d x = x^{2} + 10 x + 50 ln ∣ x - 5 ∣ - 75

\int \frac{16 x}{x - 5} d x = 16 (x - 5 + 5 ln ∣ x - 5 ∣)

\int \frac{30}{x - 5} d x = 30 ln ∣ x - 5 ∣

= x^{2} + 10 x + 50 ln ∣ x - 5 ∣ - 75 - 16 (x - 5 + 5 ln ∣ x - 5 ∣) + 30 ln ∣ x - 5 ∣

Simplify

x^{2} + 10 x + 50 ln ∣ x - 5 ∣ - 75 - 16 (x - 5 + 5 ln ∣ x - 5 ∣) + 30 ln ∣ x - 5 ∣ : x^{2} - 6 x + 5

= x^{2} - 6 x + 5

Add a constant to the solution

= x^{2} - 6 x + 5 + C

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What is the integral of (2x^2-16x+30)/(x-5) ?
The integral of (2x^2-16x+30)/(x-5) is x^2-6x+5+C