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

The integral of (8x^3-8x+5)/(x^3-x^2) is (8x^3-8x+5}{x(x-1)}+5ln|x-1|-5/(x-1)+3ln|x|+\frac{10)/x+C

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

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

\frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} + 5 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 3 ln ∣ x ∣ + \frac{10}{x} + C

Solution steps

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

Apply Integration By Parts

: \frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} - \int - \frac{8 x ^{3} - 16 x ^{2} + 23 x - 10}{x ^{2} ( x - 1 ) ^{2}} d x

= \frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} - \int - \frac{8 x ^{3} - 16 x ^{2} + 23 x - 10}{x ^{2} ( x - 1 ) ^{2}} d x

\int - \frac{8 x ^{3} - 16 x ^{2} + 23 x - 10}{x ^{2} ( x - 1 ) ^{2}} d x = - 5 ln ∣ x - 1 ∣ + \frac{5}{x - 1} - 3 ln ∣ x ∣ - \frac{10}{x}

= \frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} - (- 5 ln ∣ x - 1 ∣ + \frac{5}{x - 1} - 3 ln ∣ x ∣ - \frac{10}{x})

Simplify

= \frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} + 5 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 3 ln ∣ x ∣ + \frac{10}{x}

Add a constant to the solution

= \frac{8 x ^{3} - 8 x + 5}{x ( x - 1 )} + 5 ln ∣ x - 1 ∣ - \frac{5}{x - 1} + 3 ln ∣ x ∣ + \frac{10}{x} + C

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What is the integral of (8x^3-8x+5)/(x^3-x^2) ?
The integral of (8x^3-8x+5)/(x^3-x^2) is (8x^3-8x+5}{x(x-1)}+5ln|x-1|-5/(x-1)+3ln|x|+\frac{10)/x+C