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

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Final answer to the problem

$\frac{-1}{25\left(x-1\right)}+\frac{-1}{25\left(x+4\right)}-\frac{2}{125}\ln\left|x-1\right|+\frac{2}{125}\ln\left|x+4\right|+C_0$
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Step-by-step Solution

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

$\frac{1}{25\left(x-1\right)^2}+\frac{1}{25\left(x+4\right)^2}+\frac{-2}{125\left(x-1\right)}+\frac{2}{125\left(x+4\right)}$

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$\frac{1}{25\left(x-1\right)^2}+\frac{1}{25\left(x+4\right)^2}+\frac{-2}{125\left(x-1\right)}+\frac{2}{125\left(x+4\right)}$

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Unlock the first 3 steps of this solution

Learn how to solve problems step by step online. Find the integral int(1/((x-1)^2(x+4)^2))dx. Rewrite the fraction \frac{1}{\left(x-1\right)^2\left(x+4\right)^2} in 4 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{1}{25\left(x-1\right)^2}+\frac{1}{25\left(x+4\right)^2}+\frac{-2}{125\left(x-1\right)}+\frac{2}{125\left(x+4\right)}\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{1}{25\left(x-1\right)^2}dx results in: \frac{-1}{25\left(x-1\right)}. The integral \int\frac{1}{25\left(x+4\right)^2}dx results in: \frac{-1}{25\left(x+4\right)}.

Final answer to the problem

$\frac{-1}{25\left(x-1\right)}+\frac{-1}{25\left(x+4\right)}-\frac{2}{125}\ln\left|x-1\right|+\frac{2}{125}\ln\left|x+4\right|+C_0$

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Function Plot

Plotting: $\frac{-1}{25\left(x-1\right)}+\frac{-1}{25\left(x+4\right)}-\frac{2}{125}\ln\left(x-1\right)+\frac{2}{125}\ln\left(x+4\right)+C_0$

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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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