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

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

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

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

2 (- \frac{x ^{2}}{x - 3} + 2 (x - 3 + 3 ln ∣ x - 3 ∣)) - \frac{1}{x - 3} + C

Solution steps

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

Expand

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

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

\int \frac{2 x ^{2}}{( x - 3 ) ^{2}} d x = 2 (- \frac{x ^{2}}{x - 3} + 2 (x - 3 + 3 ln ∣ x - 3 ∣))

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

= 2 (- \frac{x ^{2}}{x - 3} + 2 (x - 3 + 3 ln ∣ x - 3 ∣)) - \frac{1}{x - 3}

Add a constant to the solution

= 2 (- \frac{x ^{2}}{x - 3} + 2 (x - 3 + 3 ln ∣ x - 3 ∣)) - \frac{1}{x - 3} + C

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