Find the integral $\int\frac{x^2+1}{x^3-6x^2+12x-8}dx$

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Final answer to the problem

$\frac{-x^2+7-4x+2\left(x-2\right)^{2}\ln\left|x-2\right|}{2\left(x-2\right)^{2}}+C_0$
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Step-by-step Solution

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  • Integrate by parts
  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
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1

We can factor the polynomial $x^3-6x^2+12x-8$ 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 $-8$

$1, 2, 4, 8$

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$1, 2, 4, 8$

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Learn how to solve problems step by step online. Find the integral int((x^2+1)/(x^3-6x^212x+-8))dx. We can factor the polynomial x^3-6x^2+12x-8 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 -8. Next, list all divisors of the leading coefficient a_n, which equals 1. The possible roots \pm\frac{p}{q} of the polynomial x^3-6x^2+12x-8 will then be. Trying all possible roots, we found that 2 is a root of the polynomial. When we evaluate it in the polynomial, it gives us 0 as a result.

Final answer to the problem

$\frac{-x^2+7-4x+2\left(x-2\right)^{2}\ln\left|x-2\right|}{2\left(x-2\right)^{2}}+C_0$

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Function Plot

Plotting: $\frac{-x^2+7-4x+2\left(x-2\right)^{2}\ln\left(x-2\right)}{2\left(x-2\right)^{2}}+C_0$

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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
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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