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

The integral of (2x^3+2x^2+1)/(x^2-x) is x^2+4x-ln|x|+5ln|x-1|+C

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

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

x^{2} + 4 x - ln ∣ x ∣ + 5 ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{2 x ^{3} + 2 x ^{2} + 1}{x ^{2} - x} : 2 x + 4 - \frac{1}{x} + \frac{5}{x - 1}

= \int 2 x + 4 - \frac{1}{x} + \frac{5}{x - 1} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 2 x d x + \int 4 d x - \int \frac{1}{x} d x + \int \frac{5}{x - 1} d x

\int 2 x d x = x^{2}

\int 4 d x = 4 x

\int \frac{1}{x} d x = ln ∣ x ∣

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

= x^{2} + 4 x - ln ∣ x ∣ + 5 ln ∣ x - 1 ∣

Add a constant to the solution

= x^{2} + 4 x - ln ∣ x ∣ + 5 ln ∣ x - 1 ∣ + C

What is the integral of (2x^3+2x^2+1)/(x^2-x) ?
The integral of (2x^3+2x^2+1)/(x^2-x) is x^2+4x-ln|x|+5ln|x-1|+C