What is the integral of (2+3x+x^2)/(x(x^2+1)) ?

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

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integral of (2+3x+x^2)/(x(x^2+1))

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

- \frac{1}{2} ln x^{2} + 1 + 2 ln ∣ x ∣ + 3 arctan (x) + C

Solution steps

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

Expand

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

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

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

\int \frac{3 x}{x ( x ^{2} + 1 )} d x = 3 arctan (x)

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

= 2 ln ∣ x ∣ - ln x^{2} + 1 + 3 arctan (x) + \frac{1}{2} ln x^{2} + 1

Simplify

2 ln ∣ x ∣ - ln x^{2} + 1 + 3 arctan (x) + \frac{1}{2} ln x^{2} + 1 : - \frac{1}{2} ln x^{2} + 1 + 2 ln ∣ x ∣ + 3 arctan (x)

= - \frac{1}{2} ln x^{2} + 1 + 2 ln ∣ x ∣ + 3 arctan (x)

Add a constant to the solution

= - \frac{1}{2} ln x^{2} + 1 + 2 ln ∣ x ∣ + 3 arctan (x) + C

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