# Evaluate. {eq}\int {{{2{x^2} - x + 4} \over {{x^3} + 4x}}} \ dx {/eq}

## Question:

Evaluate.

{eq}\int {{{2{x^2} - x + 4} \over {{x^3} + 4x}}} \ dx {/eq}

## Partial Fraction Decomposition:

In calculus mathematics, the partial fraction decomposition method is applied to break or expand the complex rational expression (mostly algebraic expression) into a sum of smaller rational expressions. Because of this method, we can easily evaluate the antiderivative of the complex rational expression. The following formula helps us to solve the given indefinite integral.

\begin{align*} \int {\dfrac{x}{{{x^2} \pm a}}\;dx} &= \dfrac{1}{2}\ln \left| {{x^2} \pm a} \right| + C\\[0.3cm] \int {\dfrac{{dx}}{{{x^2} + {a^2}}}} &={ \dfrac{1}{a}{{\tan }^{ - 1}}\left( {\dfrac{x}{a}} \right) + C} \end{align*}

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Rewrite the given indefinite integral.

\displaystyle \int {\dfrac{{2{x^2} - x + 4}}{{{x^3} + 4x}}} \;dx = \int {\dfrac{{2{x^2} - x + 4}}{{x\left(... 