Find the integral $\int\frac{x^2-3x+4}{\left(x-1\right)^3\left(x+1\right)}dx$

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Final answer to the problem

$\frac{1}{-2\left(x-1\right)^{2}}-\ln\left|x+1\right|+\ln\left|x-1\right|+\frac{1}{x-1}+C_0$
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Step-by-step Solution

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  • Integrate by partial fractions
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Rewrite the fraction $\frac{x^2-3x+4}{\left(x-1\right)^3\left(x+1\right)}$ in $4$ simpler fractions using partial fraction decomposition

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

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

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Learn how to solve problems step by step online. Find the integral int((x^2-3x+4)/((x-1)^3(x+1)))dx. Rewrite the fraction \frac{x^2-3x+4}{\left(x-1\right)^3\left(x+1\right)} in 4 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{1}{\left(x-1\right)^3}+\frac{-1}{x+1}+\frac{1}{x-1}+\frac{-1}{\left(x-1\right)^{2}}\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{1}{\left(x-1\right)^3}dx results in: \frac{1}{-2\left(x-1\right)^{2}}. The integral \int\frac{-1}{x+1}dx results in: -\ln\left(x+1\right).

Final answer to the problem

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

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Function Plot

Plotting: $\frac{1}{-2\left(x-1\right)^{2}}-\ln\left(x+1\right)+\ln\left(x-1\right)+\frac{1}{x-1}+C_0$

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5
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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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