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Evaluate the following integral: \int \frac {2x^2-x+4}{x^3+4x}dx

Question:

Evaluate the following integral: {eq}\int \frac {2x^2-x+4}{x^3+4x}dx {/eq}

Indefinite Integral:

The indefinite integral that is solved with the substitution method, the simplification and the splitting of the terms methods, etc are one of the few easy methods to solve the definite or the indefinite integrals.

Answer and Explanation:


The indefinite integral that is given here is:

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

So we can also write it as;

{eq}\int \frac{2x^2-x+4}{x\left(x^2+4\right)}dx\\ =\int \frac{x-1}{x^2+4}+\frac{1}{x}dx\\ =\int \frac{x}{x^2+4}dx-\int \frac{1}{x^2+4}dx+\frac{1}{x}dx\\ =\frac{1}{2}\ln \left|x^2+4\right|-\frac{1}{2}\arctan \left(\frac{x}{2}\right)+\ln \left|x\right|+C~~~~~~~~~~~~~~~~~~~~~~~~\left [ \because \int \frac{1}{x}dx=\ln \left(\left|x\right|\right)+c, \int \frac{1}{u^2+1}du=\arctan \left(u\right)+c \right ]\\ {/eq}

So this is the required result of the indefinite integral after simplifying and applying the integral formulas.


Learn more about this topic:

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Indefinite Integral: Definition, Rules & Examples

from Calculus: Tutoring Solution

Chapter 7 / Lesson 14
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