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

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

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

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

ln ∣ x + 1 ∣ + \frac{1}{2} arctan (\frac{x}{2}) + C

Solution steps

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

Take the partial fraction of

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

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

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

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

= ln ∣ x + 1 ∣ + \frac{1}{2} arctan (\frac{x}{2})

Add a constant to the solution

= ln ∣ x + 1 ∣ + \frac{1}{2} arctan (\frac{x}{2}) + C

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