What is the integral of (x^2-24x)/((x-12)^2) ?

The integral of (x^2-24x)/((x-12)^2) is -(x^2)/(x-12)+2x-24+(288)/(x-12)+C

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integral of (x^2-24x)/((x-12)^2)

\int \frac{x ^{2} - 24 x}{( x - 12 ) ^{2}} d x

- \frac{x ^{2}}{x - 12} + 2 x - 24 + \frac{288}{x - 12} + C

Solution steps

\int \frac{x ^{2} - 24 x}{( x - 12 ) ^{2}} d x

Expand

\frac{x ^{2} - 24 x}{( x - 12 ) ^{2}} : \frac{x ^{2}}{( x - 12 ) ^{2}} - \frac{24 x}{( x - 12 ) ^{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 ^{2}}{( x - 12 ) ^{2}} d x - \int \frac{24 x}{( x - 12 ) ^{2}} d x

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

\int \frac{24 x}{( x - 12 ) ^{2}} d x = 24 (ln ∣ x - 12 ∣ - \frac{12}{x - 12})

= - \frac{x ^{2}}{x - 12} + 2 (x - 12 + 12 ln ∣ x - 12 ∣) - 24 (ln ∣ x - 12 ∣ - \frac{12}{x - 12})

Simplify

- \frac{x ^{2}}{x - 12} + 2 (x - 12 + 12 ln ∣ x - 12 ∣) - 24 (ln ∣ x - 12 ∣ - \frac{12}{x - 12}) : - \frac{x ^{2}}{x - 12} + 2 x - 24 + \frac{288}{x - 12}

= - \frac{x ^{2}}{x - 12} + 2 x - 24 + \frac{288}{x - 12}

Add a constant to the solution

= - \frac{x ^{2}}{x - 12} + 2 x - 24 + \frac{288}{x - 12} + C

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What is the integral of (x^2-24x)/((x-12)^2) ?
The integral of (x^2-24x)/((x-12)^2) is -(x^2)/(x-12)+2x-24+(288)/(x-12)+C