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Step-by-step Solution

Integral of $\frac{x-2}{x\left(x^2+5x-14\right)}$ with respect to x

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Answer

$-\frac{1}{7}\ln\left|x+7\right|+\frac{1}{7}\ln\left|x\right|+C_0$

Step-by-step explanation

Problem to solve:

$\int\frac{x-2}{x\cdot\left(x^2+5x-14\right)}dx$
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Factor the trinomial $\left(x^2+5x-14\right)$ finding two numbers that multiply to form $-14$ and added form $5$

$\begin{matrix}\left(-2\right)\left(7\right)=-14\\ \left(-2\right)+\left(7\right)=5\end{matrix}$
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Thus

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

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Answer

$-\frac{1}{7}\ln\left|x+7\right|+\frac{1}{7}\ln\left|x\right|+C_0$

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