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

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

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integral of ((3x^4-2x^2+2))/(x^3(x-1))

\int \frac{( 3 x ^{4} - 2 x ^{2} + 2 )}{x ^{3} ( x - 1 )} d x

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

Solution steps

\int \frac{3 x ^{4} - 2 x ^{2} + 2}{x ^{3} ( x - 1 )} d x

Expand

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

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 ^{4}}{x ^{3} ( x - 1 )} d x - \int \frac{2 x ^{2}}{x ^{3} ( x - 1 )} d x + \int \frac{2}{x ^{3} ( x - 1 )} d x

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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