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

The integral of (x^3+2x)/(3x^2+5) is 1/6 x^2+5/18+1/18 ln|3x^2+5|+C

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

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

\frac{1}{6} x^{2} + \frac{5}{18} + \frac{1}{18} ln 3 x^{2} + 5 + C

Solution steps

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

Expand

\frac{x ^{3} + 2 x}{3 x ^{2} + 5} : \frac{x ^{3}}{3 x ^{2} + 5} + \frac{2 x}{3 x ^{2} + 5}

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

\int \frac{x ^{3}}{3 x ^{2} + 5} d x = \frac{1}{18} (3 x^{2} + 5 - 5 ln 3 x^{2} + 5)

\int \frac{2 x}{3 x ^{2} + 5} d x = \frac{1}{3} ln 3 x^{2} + 5

= \frac{1}{18} (3 x^{2} + 5 - 5 ln 3 x^{2} + 5) + \frac{1}{3} ln 3 x^{2} + 5

Simplify

\frac{1}{18} (3 x^{2} + 5 - 5 ln 3 x^{2} + 5) + \frac{1}{3} ln 3 x^{2} + 5 : \frac{1}{6} x^{2} + \frac{5}{18} + \frac{1}{18} ln 3 x^{2} + 5

= \frac{1}{6} x^{2} + \frac{5}{18} + \frac{1}{18} ln 3 x^{2} + 5

Add a constant to the solution

= \frac{1}{6} x^{2} + \frac{5}{18} + \frac{1}{18} ln 3 x^{2} + 5 + C

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What is the integral of (x^3+2x)/(3x^2+5) ?
The integral of (x^3+2x)/(3x^2+5) is 1/6 x^2+5/18+1/18 ln|3x^2+5|+C