What is the integral of (x^2+1)/((x-8)(x-7)^2) ?

The integral of (x^2+1)/((x-8)(x-7)^2) is 65ln|x-8|-64ln|x-7|+(50)/(x-7)+C

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integral of (x^2+1)/((x-8)(x-7)^2)

\int \frac{x ^{2} + 1}{( x - 8 ) ( x - 7 ) ^{2}} d x

65 ln ∣ x - 8 ∣ - 64 ln ∣ x - 7 ∣ + \frac{50}{x - 7} + C

Solution steps

\int \frac{x ^{2} + 1}{( x - 8 ) ( x - 7 ) ^{2}} d x

Take the partial fraction of

\frac{x ^{2} + 1}{( x - 8 ) ( x - 7 ) ^{2}} : \frac{65}{x - 8} - \frac{64}{x - 7} - \frac{50}{( x - 7 ) ^{2}}

= \int \frac{65}{x - 8} - \frac{64}{x - 7} - \frac{50}{( 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{65}{x - 8} d x - \int \frac{64}{x - 7} d x - \int \frac{50}{( x - 7 ) ^{2}} d x

\int \frac{65}{x - 8} d x = 65 ln ∣ x - 8 ∣

\int \frac{64}{x - 7} d x = 64 ln ∣ x - 7 ∣

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

= 65 ln ∣ x - 8 ∣ - 64 ln ∣ x - 7 ∣ - (- \frac{50}{x - 7})

Simplify

= 65 ln ∣ x - 8 ∣ - 64 ln ∣ x - 7 ∣ + \frac{50}{x - 7}

Add a constant to the solution

= 65 ln ∣ x - 8 ∣ - 64 ln ∣ x - 7 ∣ + \frac{50}{x - 7} + C

What is the integral of (x^2+1)/((x-8)(x-7)^2) ?
The integral of (x^2+1)/((x-8)(x-7)^2) is 65ln|x-8|-64ln|x-7|+(50)/(x-7)+C