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

The integral of (x^3+2x^2+3x-2)/(x-2) is 4x^2+7x+20ln|x-2|+1/3 (x-2)^3-30+C

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

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

4 x^{2} + 7 x + 20 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 30 + C

Solution steps

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

Expand

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

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

\int \frac{2 x ^{2}}{x - 2} d x = x^{2} + 4 x + 8 ln ∣ x - 2 ∣ - 12

\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 ∣

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

Simplify

3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 + x^{2} + 4 x + 8 ln ∣ x - 2 ∣ - 12 + 3 (x - 2 + 2 ln ∣ x - 2 ∣) - 2 ln ∣ x - 2 ∣ : 4 x^{2} + 7 x + 20 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 30

= 4 x^{2} + 7 x + 20 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 30

Add a constant to the solution

= 4 x^{2} + 7 x + 20 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 30 + C

x^{\frac{11}{2}}, 1, x = 1.4

\int \frac{2}{x} - \frac{4}{x ^{3}} d x

3 sin^{2} (t)

\int \frac{4 e ^{x}}{e ^{2 x} + 8 e ^{x} + 16} d x

\frac{\partial}{\partial x} (- x e^{\frac{x}{y}})

What is the integral of (x^3+2x^2+3x-2)/(x-2) ?
The integral of (x^3+2x^2+3x-2)/(x-2) is 4x^2+7x+20ln|x-2|+1/3 (x-2)^3-30+C