What is the integral of (8x^3-7x+5)/(x^3-x^2) ?

The integral of (8x^3-7x+5)/(x^3-x^2) is (8x^2)/(x-1)+6ln|x-1|-8/(x-1)+2ln|x|+5/x+C

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integral of (8x^3-7x+5)/(x^3-x^2)

\int \frac{8 x ^{3} - 7 x + 5}{x ^{3} - x ^{2}} d x

\frac{8 x ^{2}}{x - 1} + 6 ln ∣ x - 1 ∣ - \frac{8}{x - 1} + 2 ln ∣ x ∣ + \frac{5}{x} + C

Solution steps

\int \frac{8 x ^{3} - 7 x + 5}{x ^{3} - x ^{2}} d x

Expand

\frac{8 x ^{3} - 7 x + 5}{x ^{3} - x ^{2}} : \frac{8 x ^{3}}{x ^{3} - x ^{2}} - \frac{7 x}{x ^{3} - x ^{2}} + \frac{5}{x ^{3} - x ^{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{8 x ^{3}}{x ^{3} - x ^{2}} d x - \int \frac{7 x}{x ^{3} - x ^{2}} d x + \int \frac{5}{x ^{3} - x ^{2}} d x

\int \frac{8 x ^{3}}{x ^{3} - x ^{2}} d x = 8 (\frac{x ^{2}}{x - 1} + ln ∣ x - 1 ∣ - \frac{1}{x - 1})

\int \frac{7 x}{x ^{3} - x ^{2}} d x = 7 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

\int \frac{5}{x ^{3} - x ^{2}} d x = 5 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣)

= 8 (\frac{x ^{2}}{x - 1} + ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - 7 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) + 5 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣)

Simplify

8 (\frac{x ^{2}}{x - 1} + ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - 7 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) + 5 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣) : \frac{8 x ^{2}}{x - 1} + 6 ln ∣ x - 1 ∣ - \frac{8}{x - 1} + 2 ln ∣ x ∣ + \frac{5}{x}

= \frac{8 x ^{2}}{x - 1} + 6 ln ∣ x - 1 ∣ - \frac{8}{x - 1} + 2 ln ∣ x ∣ + \frac{5}{x}

Add a constant to the solution

= \frac{8 x ^{2}}{x - 1} + 6 ln ∣ x - 1 ∣ - \frac{8}{x - 1} + 2 ln ∣ x ∣ + \frac{5}{x} + C

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What is the integral of (8x^3-7x+5)/(x^3-x^2) ?
The integral of (8x^3-7x+5)/(x^3-x^2) is (8x^2)/(x-1)+6ln|x-1|-8/(x-1)+2ln|x|+5/x+C