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

The integral of (x^2+1)/((x-2)^2) is -(x^2+1)/(x-2)+2(x-2+2ln|x-2|)+C

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

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

- \frac{x ^{2} + 1}{x - 2} + 2 (x - 2 + 2 ln ∣ x - 2 ∣) + C

Solution steps

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

Apply Integration By Parts

= - \frac{x ^{2} + 1}{x - 2} - \int - \frac{2 x}{x - 2} d x

\int - \frac{2 x}{x - 2} d x = - 2 (x - 2 + 2 ln ∣ x - 2 ∣)

= - \frac{x ^{2} + 1}{x - 2} - (- 2 (x - 2 + 2 ln ∣ x - 2 ∣))

Simplify

= - \frac{x ^{2} + 1}{x - 2} + 2 (x - 2 + 2 ln ∣ x - 2 ∣)

Add a constant to the solution

= - \frac{x ^{2} + 1}{x - 2} + 2 (x - 2 + 2 ln ∣ x - 2 ∣) + C

What is the integral of (x^2+1)/((x-2)^2) ?
The integral of (x^2+1)/((x-2)^2) is -(x^2+1)/(x-2)+2(x-2+2ln|x-2|)+C