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

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

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

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

3 arctan (x) + ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

= 3 arctan (x) + ln ∣ x ∣

Add a constant to the solution

= 3 arctan (x) + ln ∣ x ∣ + C

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