What is the integral of (x^2+1)/((x-10)(x-9)^2) ?

The integral of (x^2+1)/((x-10)(x-9)^2) is 101ln|x-10|-100ln|x-9|+(82)/(x-9)+C

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integral of (x^2+1)/((x-10)(x-9)^2)

\int \frac{x ^{2} + 1}{( x - 10 ) ( x - 9 ) ^{2}} d x

101 ln ∣ x - 10 ∣ - 100 ln ∣ x - 9 ∣ + \frac{82}{x - 9} + C

Solution steps

\int \frac{x ^{2} + 1}{( x - 10 ) ( x - 9 ) ^{2}} d x

Take the partial fraction of

\frac{x ^{2} + 1}{( x - 10 ) ( x - 9 ) ^{2}} : \frac{101}{x - 10} - \frac{100}{x - 9} - \frac{82}{( x - 9 ) ^{2}}

= \int \frac{101}{x - 10} - \frac{100}{x - 9} - \frac{82}{( x - 9 ) ^{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{101}{x - 10} d x - \int \frac{100}{x - 9} d x - \int \frac{82}{( x - 9 ) ^{2}} d x

\int \frac{101}{x - 10} d x = 101 ln ∣ x - 10 ∣

\int \frac{100}{x - 9} d x = 100 ln ∣ x - 9 ∣

\int \frac{82}{( x - 9 ) ^{2}} d x = - \frac{82}{x - 9}

= 101 ln ∣ x - 10 ∣ - 100 ln ∣ x - 9 ∣ - (- \frac{82}{x - 9})

Simplify

= 101 ln ∣ x - 10 ∣ - 100 ln ∣ x - 9 ∣ + \frac{82}{x - 9}

Add a constant to the solution

= 101 ln ∣ x - 10 ∣ - 100 ln ∣ x - 9 ∣ + \frac{82}{x - 9} + C

What is the integral of (x^2+1)/((x-10)(x-9)^2) ?
The integral of (x^2+1)/((x-10)(x-9)^2) is 101ln|x-10|-100ln|x-9|+(82)/(x-9)+C