What is the integral of (5x^2-33x+54)/((x-2)(x-4)^2) ?

The integral of (5x^2-33x+54)/((x-2)(x-4)^2) is 2ln|x-2|+3ln|x-4|-1/(x-4)+C

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integral of (5x^2-33x+54)/((x-2)(x-4)^2)

\int \frac{5 x ^{2} - 33 x + 54}{( x - 2 ) ( x - 4 ) ^{2}} d x

2 ln ∣ x - 2 ∣ + 3 ln ∣ x - 4 ∣ - \frac{1}{x - 4} + C

Solution steps

\int \frac{5 x ^{2} - 33 x + 54}{( x - 2 ) ( x - 4 ) ^{2}} d x

Take the partial fraction of

\frac{5 x ^{2} - 33 x + 54}{( x - 2 ) ( x - 4 ) ^{2}} : \frac{2}{x - 2} + \frac{3}{x - 4} + \frac{1}{( x - 4 ) ^{2}}

= \int \frac{2}{x - 2} + \frac{3}{x - 4} + \frac{1}{( x - 4 ) ^{2}} 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} d x + \int \frac{3}{x - 4} d x + \int \frac{1}{( x - 4 ) ^{2}} d x

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

\int \frac{3}{x - 4} d x = 3 ln ∣ x - 4 ∣

\int \frac{1}{( x - 4 ) ^{2}} d x = - \frac{1}{x - 4}

= 2 ln ∣ x - 2 ∣ + 3 ln ∣ x - 4 ∣ - \frac{1}{x - 4}

Add a constant to the solution

= 2 ln ∣ x - 2 ∣ + 3 ln ∣ x - 4 ∣ - \frac{1}{x - 4} + C

What is the integral of (5x^2-33x+54)/((x-2)(x-4)^2) ?
The integral of (5x^2-33x+54)/((x-2)(x-4)^2) is 2ln|x-2|+3ln|x-4|-1/(x-4)+C