What is the integral of (x^2+1)/((x-7)(x-6)^2) ?

The integral of (x^2+1)/((x-7)(x-6)^2) is 50ln|x-7|-49ln|x-6|+(37)/(x-6)+C

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integral of (x^2+1)/((x-7)(x-6)^2)

\int \frac{x ^{2} + 1}{( x - 7 ) ( x - 6 ) ^{2}} d x

50 ln ∣ x - 7 ∣ - 49 ln ∣ x - 6 ∣ + \frac{37}{x - 6} + C

Solution steps

\int \frac{x ^{2} + 1}{( x - 7 ) ( x - 6 ) ^{2}} d x

Take the partial fraction of

\frac{x ^{2} + 1}{( x - 7 ) ( x - 6 ) ^{2}} : \frac{50}{x - 7} - \frac{49}{x - 6} - \frac{37}{( x - 6 ) ^{2}}

= \int \frac{50}{x - 7} - \frac{49}{x - 6} - \frac{37}{( 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{50}{x - 7} d x - \int \frac{49}{x - 6} d x - \int \frac{37}{( x - 6 ) ^{2}} d x

\int \frac{50}{x - 7} d x = 50 ln ∣ x - 7 ∣

\int \frac{49}{x - 6} d x = 49 ln ∣ x - 6 ∣

\int \frac{37}{( x - 6 ) ^{2}} d x = - \frac{37}{x - 6}

= 50 ln ∣ x - 7 ∣ - 49 ln ∣ x - 6 ∣ - (- \frac{37}{x - 6})

Simplify

= 50 ln ∣ x - 7 ∣ - 49 ln ∣ x - 6 ∣ + \frac{37}{x - 6}

Add a constant to the solution

= 50 ln ∣ x - 7 ∣ - 49 ln ∣ x - 6 ∣ + \frac{37}{x - 6} + C

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What is the integral of (x^2+1)/((x-7)(x-6)^2) ?
The integral of (x^2+1)/((x-7)(x-6)^2) is 50ln|x-7|-49ln|x-6|+(37)/(x-6)+C