What is the integral of (x^5)/((x-2)^2) ?

The integral of (x^5)/((x-2)^2) is (x^4)/4+(4x^3)/3+6x^2+32x-(80x-128)/(x-2)+80ln|x-2|+C

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integral of (x^5)/((x-2)^2)

\int \frac{x ^{5}}{( x - 2 ) ^{2}} d x

\frac{x ^{4}}{4} + \frac{4 x ^{3}}{3} + 6 x^{2} + 32 x - \frac{80 x - 128}{x - 2} + 80 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x ^{5}}{( x - 2 ) ^{2}} d x

Long division

\frac{x ^{5}}{( x - 2 ) ^{2}} : x^{3} + 4 x^{2} + 12 x + 32 + \frac{80 x - 128}{x ^{2} - 4 x + 4}

= \int x^{3} + 4 x^{2} + 12 x + 32 + \frac{80 x - 128}{x ^{2} - 4 x + 4} 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 x^{3} d x + \int 4 x^{2} d x + \int 12 x d x + \int 32 d x + \int \frac{80 x - 128}{x ^{2} - 4 x + 4} d x

\int x^{3} d x = \frac{x ^{4}}{4}

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

\int 12 x d x = 6 x^{2}

\int 32 d x = 32 x

\int \frac{80 x - 128}{x ^{2} - 4 x + 4} d x = - \frac{80 x - 128}{x - 2} + 80 ln ∣ x - 2 ∣

= \frac{x ^{4}}{4} + \frac{4 x ^{3}}{3} + 6 x^{2} + 32 x - \frac{80 x - 128}{x - 2} + 80 ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{x ^{4}}{4} + \frac{4 x ^{3}}{3} + 6 x^{2} + 32 x - \frac{80 x - 128}{x - 2} + 80 ln ∣ x - 2 ∣ + C

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What is the integral of (x^5)/((x-2)^2) ?
The integral of (x^5)/((x-2)^2) is (x^4)/4+(4x^3)/3+6x^2+32x-(80x-128)/(x-2)+80ln|x-2|+C