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Step-by-step Solution

Integral of $\frac{1}{x^3+2x^2+x}$ with respect to x

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Answer

$\ln\left|x\right|-\frac{3}{2}\ln\left|x+1\right|+C_0$

Step-by-step explanation

Problem to solve:

$\int\left(\frac{dx}{x^3+2x^2+x}\right)$
1

We can factor the polynomial $x^3+2x^2+x$ using synthetic division (Ruffini's rule). We found that $-1$ is a root of the polynomial

${\left(-1\right)}^3+2{\left(-1\right)}^2-1=0$
2

Let's divide the polynomial by $x+1$ using synthetic division. First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $-1$. Add the result to the second coefficient and then multiply this by $-1$ and so on

$\left|\begin{array}{c}1 & 2 & 1 & 0 \\ & -1 & -1 & 0 \\ 1 & 1 & 0 & 0\end{array}\right|-1$

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Answer

$\ln\left|x\right|-\frac{3}{2}\ln\left|x+1\right|+C_0$

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