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

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

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

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

x - \frac{3}{x} + 4 ln ∣ x + 2 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \int 1 + \frac{3}{x ^{2}} + \frac{4}{x + 2} 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 1 d x + \int \frac{3}{x ^{2}} d x + \int \frac{4}{x + 2} d x

\int 1 d x = x

\int \frac{3}{x ^{2}} d x = - \frac{3}{x}

\int \frac{4}{x + 2} d x = 4 ln ∣ x + 2 ∣

= x - \frac{3}{x} + 4 ln ∣ x + 2 ∣

Add a constant to the solution

= x - \frac{3}{x} + 4 ln ∣ x + 2 ∣ + C

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