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

The integral of (2x^3-3x^2-3x+2)/(x-2) is 6x^2-15x+2/3 (x-2)^3-3/2 (x-2)^2+6+C

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

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

6 x^{2} - 15 x + \frac{2}{3} (x - 2)^{3} - \frac{3}{2} (x - 2)^{2} + 6 + C

Solution steps

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

Expand

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

\int \frac{2 x ^{3}}{x - 2} d x = 2 (3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12)

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

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

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

= 2 (3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12) - 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) - 3 (x - 2 + 2 ln ∣ x - 2 ∣) + 2 ln ∣ x - 2 ∣

Simplify

2 (3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12) - 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) - 3 (x - 2 + 2 ln ∣ x - 2 ∣) + 2 ln ∣ x - 2 ∣ : 6 x^{2} - 15 x + \frac{2}{3} (x - 2)^{3} - \frac{3}{2} (x - 2)^{2} + 6

= 6 x^{2} - 15 x + \frac{2}{3} (x - 2)^{3} - \frac{3}{2} (x - 2)^{2} + 6

Add a constant to the solution

= 6 x^{2} - 15 x + \frac{2}{3} (x - 2)^{3} - \frac{3}{2} (x - 2)^{2} + 6 + C

\int_{0}^{2} x - 1 d x

x^{^{''}} + 2 x = 0, x (0) = 1, x (1) = 0

13440 (2 x + 5)^{3}

y = x^{2} - 2 x - 2 P (y, x), (2, - 2)

x \to 0 lim (\frac{x}{sin ( 2 x )})

What is the integral of (2x^3-3x^2-3x+2)/(x-2) ?
The integral of (2x^3-3x^2-3x+2)/(x-2) is 6x^2-15x+2/3 (x-2)^3-3/2 (x-2)^2+6+C