What is the integral of (x^2)/(x^3+5x^2+8x+4) ?

The integral of (x^2)/(x^3+5x^2+8x+4) is ln|x+1|+4/(x+2)+C

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integral of (x^2)/(x^3+5x^2+8x+4)

\int \frac{x ^{2}}{x ^{3} + 5 x ^{2} + 8 x + 4} d x

ln ∣ x + 1 ∣ + \frac{4}{x + 2} + C

Solution steps

\int \frac{x ^{2}}{x ^{3} + 5 x ^{2} + 8 x + 4} d x

Take the partial fraction of

\frac{x ^{2}}{x ^{3} + 5 x ^{2} + 8 x + 4} : \frac{1}{x + 1} - \frac{4}{( x + 2 ) ^{2}}

= \int \frac{1}{x + 1} - \frac{4}{( x + 2 ) ^{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 \frac{1}{x + 1} d x - \int \frac{4}{( x + 2 ) ^{2}} d x

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

\int \frac{4}{( x + 2 ) ^{2}} d x = - \frac{4}{x + 2}

= ln ∣ x + 1 ∣ - (- \frac{4}{x + 2})

Simplify

= ln ∣ x + 1 ∣ + \frac{4}{x + 2}

Add a constant to the solution

= ln ∣ x + 1 ∣ + \frac{4}{x + 2} + C

What is the integral of (x^2)/(x^3+5x^2+8x+4) ?
The integral of (x^2)/(x^3+5x^2+8x+4) is ln|x+1|+4/(x+2)+C