What is the integral of (x^2)/((x-6)(x-7)^2) ?

The integral of (x^2)/((x-6)(x-7)^2) is 36ln|x-6|-35ln|x-7|-(49)/(x-7)+C

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integral of (x^2)/((x-6)(x-7)^2)

\int \frac{x ^{2}}{( x - 6 ) ( x - 7 ) ^{2}} d x

36 ln ∣ x - 6 ∣ - 35 ln ∣ x - 7 ∣ - \frac{49}{x - 7} + C

Solution steps

\int \frac{x ^{2}}{( x - 6 ) ( x - 7 ) ^{2}} d x

Take the partial fraction of

\frac{x ^{2}}{( x - 6 ) ( x - 7 ) ^{2}} : \frac{36}{x - 6} - \frac{35}{x - 7} + \frac{49}{( x - 7 ) ^{2}}

= \int \frac{36}{x - 6} - \frac{35}{x - 7} + \frac{49}{( x - 7 ) ^{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{36}{x - 6} d x - \int \frac{35}{x - 7} d x + \int \frac{49}{( x - 7 ) ^{2}} d x

\int \frac{36}{x - 6} d x = 36 ln ∣ x - 6 ∣

\int \frac{35}{x - 7} d x = 35 ln ∣ x - 7 ∣

\int \frac{49}{( x - 7 ) ^{2}} d x = - \frac{49}{x - 7}

= 36 ln ∣ x - 6 ∣ - 35 ln ∣ x - 7 ∣ - \frac{49}{x - 7}

Add a constant to the solution

= 36 ln ∣ x - 6 ∣ - 35 ln ∣ x - 7 ∣ - \frac{49}{x - 7} + C

What is the integral of (x^2)/((x-6)(x-7)^2) ?
The integral of (x^2)/((x-6)(x-7)^2) is 36ln|x-6|-35ln|x-7|-(49)/(x-7)+C