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

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

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

\int \frac{x ^{2} + 3 x + 2}{x ( x ^{2} + 2 x + 2 )} d x

ln ∣ x ∣ + arctan (x + 1) + C

Solution steps

\int \frac{x ^{2} + 3 x + 2}{x ( x ^{2} + 2 x + 2 )} d x

Take the partial fraction of

\frac{x ^{2} + 3 x + 2}{x ( x ^{2} + 2 x + 2 )} : \frac{1}{x} + \frac{1}{x ^{2} + 2 x + 2}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + arctan (x + 1)

Add a constant to the solution

= ln ∣ x ∣ + arctan (x + 1) + C

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