What is the integral of (2x^2-5x+4)/((x-2)^3) ?

The integral of (2x^2-5x+4)/((x-2)^3) is -(x^2)/((x-2)^2)+2ln|x-2|+1/(x-2)+3/((x-2)^2)+C

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integral of (2x^2-5x+4)/((x-2)^3)

\int \frac{2 x ^{2} - 5 x + 4}{( x - 2 ) ^{3}} d x

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

Solution steps

\int \frac{2 x ^{2} - 5 x + 4}{( x - 2 ) ^{3}} d x

Expand

\frac{2 x ^{2} - 5 x + 4}{( x - 2 ) ^{3}} : \frac{2 x ^{2}}{( x - 2 ) ^{3}} - \frac{5 x}{( x - 2 ) ^{3}} + \frac{4}{( x - 2 ) ^{3}}

= \int \frac{2 x ^{2}}{( x - 2 ) ^{3}} - \frac{5 x}{( x - 2 ) ^{3}} + \frac{4}{( x - 2 ) ^{3}} d x

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 - 2 ) ^{3}} d x - \int \frac{5 x}{( x - 2 ) ^{3}} d x + \int \frac{4}{( x - 2 ) ^{3}} d x

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

\int \frac{5 x}{( x - 2 ) ^{3}} d x = 5 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}})

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

= - \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ - \frac{4}{x - 2} - 5 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) - \frac{2}{( x - 2 ) ^{2}}

Simplify

- \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ - \frac{4}{x - 2} - 5 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) - \frac{2}{( x - 2 ) ^{2}} : - \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ + \frac{1}{x - 2} + \frac{3}{( x - 2 ) ^{2}}

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

Add a constant to the solution

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

\int 2 x (x^{2} - 3)^{3} d x

\int x^{k} d x

(x + 2) (x^{3} - x)

\frac{d}{d x} (\frac{x ^{3}}{x})

\int \frac{1}{( 16 - x ^{2} ) ^{3}} d x

What is the integral of (2x^2-5x+4)/((x-2)^3) ?
The integral of (2x^2-5x+4)/((x-2)^3) is -(x^2)/((x-2)^2)+2ln|x-2|+1/(x-2)+3/((x-2)^2)+C