$\int \frac{x^{4}}{(x + 2) (x^{2} + 1)} d x$
How to change this improper function to Rational function.

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Solution

$\int \frac{x^{4}}{(x + 2) (x^{2} + 1)} d x$
$= \int \frac{{x^{2} (x^{2} + 1) - x (x + 2) + 2 x + 1}}{(x + 2) (x^{2} + 1)} d x$
$= \int \frac{x^{2}}{x + 2} d x - \int \frac{x}{x^{2} + 1} d x + \int \frac{2 x + 2}{(x + 2) (x^{2} + 1)} d x$
$= \int \frac{x^{2}}{x + 2} d x - \int \frac{x}{x^{2} + 1} d x + 2 \int \frac{(x + 2) - 3 / 2}{(x + 2) (x^{2} + 1)} d x$
$= \int \frac{x^{2}}{x + 2} d x - \int \frac{(x + 2)}{x^{2} + 1} d x - 3 \int \frac{d x}{(x + 2) (x^{2} + 1)}$

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