# Step-by-step Solution

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
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

## Step-by-step explanation

Problem to solve:

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

Learn how to solve integrals by partial fraction expansion problems step by step online.

$1$

Learn how to solve integrals by partial fraction expansion problems step by step online. Integral of (2x^2+3)/(x^3-2x^2+x) with respect to x. We can factor the polynomial x^3-2x^2+x using the rational root theorem, which guarantees that for a polynomial of the form a_nx^n+a_{n-1}x^{n-1}+\dots+a_0 there is a rational root of the form \pm\frac{p}{q}, where p belongs to the divisors of the constant term a_0, and q belongs to the divisors of the leading coefficient a_n. List all divisors p of the constant term a_0, which equals 0. Next, list all divisors of the leading coefficient a_n, which equals 1. The possible roots \pm\frac{p}{q} of the polynomial x^3-2x^2+x will then be. 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.

$\frac{-5}{x-1}+3\ln\left|x\right|-\ln\left|x-1\right|+C_0$

### Problem Analysis

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

### Main topic:

Integrals by partial fraction expansion

~ 0.17 seconds