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

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

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

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

x^{2} + 2 - \frac{5}{2} ln x^{2} + 2 + C

Solution steps

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

Expand

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

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

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

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

= x^{2} + 2 - 2 ln x^{2} + 2 - \frac{1}{2} ln x^{2} + 2

Add similar elements:

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

= x^{2} + 2 - \frac{5}{2} ln x^{2} + 2

Add a constant to the solution

= x^{2} + 2 - \frac{5}{2} ln x^{2} + 2 + C

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