# Step-by-step Solution

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

Go
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
2

e
π
ln
log
lim
d/dx
Dx
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

### Videos

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

## 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

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

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

### Main topic:

Integrals of Rational Functions

~ 2.72 seconds

### Struggling with math?

Access detailed step by step solutions to millions of problems, growing every day!