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Step-by-step Solution

Integral of $\frac{7x^2-x+16}{x^3+4x}$ with respect to x

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Answer

$-\frac{1}{2}arctan\left(\frac{1}{2}x\right)+4\ln\left|x\right|+\frac{3}{2}\ln\left|x^{2}+4\right|+C_0$

Step-by-step explanation

Problem to solve:

$\int\:\frac{7x^2-x+16}{x^3+4x}dx$
1

Split the fraction $\frac{7x^2+-x+16}{x^3+4x}$ in two terms with common denominator ($x^3+4x$)

$\int\left(\frac{7x^2}{x^3+4x}+\frac{-x+16}{x^3+4x}\right)dx$
2

Split the fraction $\frac{-x+16}{x^3+4x}$ in two terms with common denominator $x^3+4x$

$\int\left(\frac{7x^2}{x^3+4x}+\frac{-x}{x^3+4x}+\frac{16}{x^3+4x}\right)dx$

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Answer

$-\frac{1}{2}arctan\left(\frac{1}{2}x\right)+4\ln\left|x\right|+\frac{3}{2}\ln\left|x^{2}+4\right|+C_0$