What is the integral of (2x^2+1)/((x-2)^3) ?

The integral of (2x^2+1)/((x-2)^3) is -(2x^2+1)/(2(x-2)^2)+2(ln|x-2|-2/(x-2))+C

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integral of (2x^2+1)/((x-2)^3)

\int \frac{2 x ^{2} + 1}{( x - 2 ) ^{3}} d x

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

Solution steps

\int \frac{2 x ^{2} + 1}{( x - 2 ) ^{3}} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (2x^2+1)/((x-2)^3) ?
The integral of (2x^2+1)/((x-2)^3) is -(2x^2+1)/(2(x-2)^2)+2(ln|x-2|-2/(x-2))+C