What is the integral of (5x^2-x+3)/(x(x^2+1)) ?

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

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integral of (5x^2-x+3)/(x(x^2+1))

\int \frac{5 x ^{2} - x + 3}{x ( x ^{2} + 1 )} d x

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

Solution steps

\int \frac{5 x ^{2} - x + 3}{x ( x ^{2} + 1 )} d x

Expand

\frac{5 x ^{2} - x + 3}{x ( x ^{2} + 1 )} : \frac{5 x ^{2}}{x ( x ^{2} + 1 )} - \frac{x}{x ( x ^{2} + 1 )} + \frac{3}{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{5 x ^{2}}{x ( x ^{2} + 1 )} d x - \int \frac{x}{x ( x ^{2} + 1 )} d x + \int \frac{3}{x ( x ^{2} + 1 )} d x

\int \frac{5 x ^{2}}{x ( x ^{2} + 1 )} d x = \frac{5}{2} ln x^{2} + 1

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

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

= \frac{5}{2} ln x^{2} + 1 - arctan (x) + 3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Simplify

\frac{5}{2} ln x^{2} + 1 - arctan (x) + 3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) : ln x^{2} + 1 - arctan (x) + 3 ln ∣ x ∣

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

Add a constant to the solution

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

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