What is the integral of (x^2)/((x-5)(x-6)^2) ?

The integral of (x^2)/((x-5)(x-6)^2) is 25ln|x-5|-24ln|x-6|-(36)/(x-6)+C

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integral of (x^2)/((x-5)(x-6)^2)

\int \frac{x ^{2}}{( x - 5 ) ( x - 6 ) ^{2}} d x

25 ln ∣ x - 5 ∣ - 24 ln ∣ x - 6 ∣ - \frac{36}{x - 6} + C

Solution steps

\int \frac{x ^{2}}{( x - 5 ) ( x - 6 ) ^{2}} d x

Take the partial fraction of

\frac{x ^{2}}{( x - 5 ) ( x - 6 ) ^{2}} : \frac{25}{x - 5} - \frac{24}{x - 6} + \frac{36}{( x - 6 ) ^{2}}

= \int \frac{25}{x - 5} - \frac{24}{x - 6} + \frac{36}{( x - 6 ) ^{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{25}{x - 5} d x - \int \frac{24}{x - 6} d x + \int \frac{36}{( x - 6 ) ^{2}} d x

\int \frac{25}{x - 5} d x = 25 ln ∣ x - 5 ∣

\int \frac{24}{x - 6} d x = 24 ln ∣ x - 6 ∣

\int \frac{36}{( x - 6 ) ^{2}} d x = - \frac{36}{x - 6}

= 25 ln ∣ x - 5 ∣ - 24 ln ∣ x - 6 ∣ - \frac{36}{x - 6}

Add a constant to the solution

= 25 ln ∣ x - 5 ∣ - 24 ln ∣ x - 6 ∣ - \frac{36}{x - 6} + C

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What is the integral of (x^2)/((x-5)(x-6)^2) ?
The integral of (x^2)/((x-5)(x-6)^2) is 25ln|x-5|-24ln|x-6|-(36)/(x-6)+C