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

The integral of (x^3)/((x-2)^2) is -(x^3)/(x-2)+3(1/2 (x-2)^2+4(x-2)+4ln|x-2|)+C

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

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

- \frac{x ^{3}}{x - 2} + 3 (\frac{1}{2} (x - 2)^{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + C

Solution steps

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

Apply Integration By Parts

: - \frac{x ^{3}}{x - 2} - \int - \frac{3 x ^{2}}{x - 2} d x

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

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

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

Simplify

- \frac{x ^{3}}{x - 2} - (- 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣)) : - \frac{x ^{3}}{x - 2} + 3 (\frac{1}{2} (x - 2)^{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣)

= - \frac{x ^{3}}{x - 2} + 3 (\frac{1}{2} (x - 2)^{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣)

Add a constant to the solution

= - \frac{x ^{3}}{x - 2} + 3 (\frac{1}{2} (x - 2)^{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + C

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What is the integral of (x^3)/((x-2)^2) ?
The integral of (x^3)/((x-2)^2) is -(x^3)/(x-2)+3(1/2 (x-2)^2+4(x-2)+4ln|x-2|)+C