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

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

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

integral

\frac{( x ^{2} - x )}{( x ^{2} + 2 )}

- 2 arctan (\frac{x}{2}) + x - \frac{1}{2} ln x^{2} + 2 + C

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

(2 x^{2})^{\frac{1}{3}}

\frac{( t - 3 )}{( t ^{2} + 7 )}

e^{2 + x}

- 2 sec^{2} (x)

sin^{7} (x)

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