What is the integral of (3x^2+4)/((x-2)^2) ?

The integral of (3x^2+4)/((x-2)^2) is 3(-(x^2)/(x-2)+2(x-2+2ln|x-2|))-4/(x-2)+C

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integral of (3x^2+4)/((x-2)^2)

\int \frac{3 x ^{2} + 4}{( x - 2 ) ^{2}} d x

3 (- \frac{x ^{2}}{x - 2} + 2 (x - 2 + 2 ln ∣ x - 2 ∣)) - \frac{4}{x - 2} + C

Solution steps

\int \frac{3 x ^{2} + 4}{( x - 2 ) ^{2}} d x

Expand

\frac{3 x ^{2} + 4}{( x - 2 ) ^{2}} : \frac{3 x ^{2}}{( x - 2 ) ^{2}} + \frac{4}{( x - 2 ) ^{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{3 x ^{2}}{( x - 2 ) ^{2}} d x + \int \frac{4}{( x - 2 ) ^{2}} d x

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

\int \frac{4}{( x - 2 ) ^{2}} d x = - \frac{4}{x - 2}

= 3 (- \frac{x ^{2}}{x - 2} + 2 (x - 2 + 2 ln ∣ x - 2 ∣)) - \frac{4}{x - 2}

Add a constant to the solution

= 3 (- \frac{x ^{2}}{x - 2} + 2 (x - 2 + 2 ln ∣ x - 2 ∣)) - \frac{4}{x - 2} + C

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What is the integral of (3x^2+4)/((x-2)^2) ?
The integral of (3x^2+4)/((x-2)^2) is 3(-(x^2)/(x-2)+2(x-2+2ln|x-2|))-4/(x-2)+C