What is the integral of (2x^3+x^2-5)/(x-1) ?

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

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integral of (2x^3+x^2-5)/(x-1)

\int \frac{2 x ^{3} + x ^{2} - 5}{x - 1} d x

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

Solution steps

\int \frac{2 x ^{3} + x ^{2} - 5}{x - 1} d x

Expand

\frac{2 x ^{3} + x ^{2} - 5}{x - 1} : \frac{2 x ^{3}}{x - 1} + \frac{x ^{2}}{x - 1} - \frac{5}{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{2 x ^{3}}{x - 1} d x + \int \frac{x ^{2}}{x - 1} d x - \int \frac{5}{x - 1} d x

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

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

\int \frac{5}{x - 1} d x = 5 ln ∣ x - 1 ∣

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

Simplify

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

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

Add a constant to the solution

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

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