What is the integral of (6x^2+8x+2)/(x^3+2x^2+x) ?

The integral of (6x^2+8x+2)/(x^3+2x^2+x) is 2ln|x|+4ln|x+1|+C

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integral of (6x^2+8x+2)/(x^3+2x^2+x)

\int \frac{6 x ^{2} + 8 x + 2}{x ^{3} + 2 x ^{2} + x} d x

2 ln ∣ x ∣ + 4 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{6 x ^{2} + 8 x + 2}{x ^{3} + 2 x ^{2} + x} d x

Take the partial fraction of

\frac{6 x ^{2} + 8 x + 2}{x ^{3} + 2 x ^{2} + x} : \frac{2}{x} + \frac{4}{x + 1}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{4}{x + 1} d x = 4 ln ∣ x + 1 ∣

= 2 ln ∣ x ∣ + 4 ln ∣ x + 1 ∣

Add a constant to the solution

= 2 ln ∣ x ∣ + 4 ln ∣ x + 1 ∣ + C

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What is the integral of (6x^2+8x+2)/(x^3+2x^2+x) ?
The integral of (6x^2+8x+2)/(x^3+2x^2+x) is 2ln|x|+4ln|x+1|+C