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

The integral of (5-x^2)/(3x^3+x) is 5ln|x|-8/3 ln|3x^2+1|+C

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

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

5 ln ∣ x ∣ - \frac{8}{3} ln 3 x^{2} + 1 + C

Solution steps

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

Expand

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

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

\int \frac{x ^{2}}{3 x ^{3} + x} d x = \frac{1}{6} ln 3 x^{2} + 1

= 5 (ln ∣ x ∣ - \frac{1}{2} ln 3 x^{2} + 1) - \frac{1}{6} ln 3 x^{2} + 1

Simplify

5 (ln ∣ x ∣ - \frac{1}{2} ln 3 x^{2} + 1) - \frac{1}{6} ln 3 x^{2} + 1 : 5 ln ∣ x ∣ - \frac{8}{3} ln 3 x^{2} + 1

= 5 ln ∣ x ∣ - \frac{8}{3} ln 3 x^{2} + 1

Add a constant to the solution

= 5 ln ∣ x ∣ - \frac{8}{3} ln 3 x^{2} + 1 + C

What is the integral of (5-x^2)/(3x^3+x) ?
The integral of (5-x^2)/(3x^3+x) is 5ln|x|-8/3 ln|3x^2+1|+C