What is the integral of (3x^2-10x-8)/(x-3) ?

The integral of (3x^2-10x-8)/(x-3) is 8x+3/2 (x-3)^2-11ln|x-3|-24+C

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integral of (3x^2-10x-8)/(x-3)

\int \frac{3 x ^{2} - 10 x - 8}{x - 3} d x

8 x + \frac{3}{2} (x - 3)^{2} - 11 ln ∣ x - 3 ∣ - 24 + C

Solution steps

\int \frac{3 x ^{2} - 10 x - 8}{x - 3} d x

Expand

\frac{3 x ^{2} - 10 x - 8}{x - 3} : \frac{3 x ^{2}}{x - 3} - \frac{10 x}{x - 3} - \frac{8}{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{3 x ^{2}}{x - 3} d x - \int \frac{10 x}{x - 3} d x - \int \frac{8}{x - 3} d x

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

\int \frac{10 x}{x - 3} d x = 10 (x - 3 + 3 ln ∣ x - 3 ∣)

\int \frac{8}{x - 3} d x = 8 ln ∣ x - 3 ∣

= 3 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣) - 10 (x - 3 + 3 ln ∣ x - 3 ∣) - 8 ln ∣ x - 3 ∣

Simplify

3 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣) - 10 (x - 3 + 3 ln ∣ x - 3 ∣) - 8 ln ∣ x - 3 ∣ : 8 x + \frac{3}{2} (x - 3)^{2} - 11 ln ∣ x - 3 ∣ - 24

= 8 x + \frac{3}{2} (x - 3)^{2} - 11 ln ∣ x - 3 ∣ - 24

Add a constant to the solution

= 8 x + \frac{3}{2} (x - 3)^{2} - 11 ln ∣ x - 3 ∣ - 24 + C

\frac{d y}{d x} = - \frac{x}{y}

\int 5 tan^{3} (2 x) sec^{5} (2 x) d x

f (g (x))

y = e^{x}, y = e^{3 x}, x = 0, x = ln (3)

x \to 3 lim (ln (3 - x))

What is the integral of (3x^2-10x-8)/(x-3) ?
The integral of (3x^2-10x-8)/(x-3) is 8x+3/2 (x-3)^2-11ln|x-3|-24+C