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# 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 

How should I solve this problem?

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

$\frac{1}{\left(x-1\right)^2\left(x+4\right)^2}=\frac{A}{\left(x-1\right)^2}+\frac{B}{\left(x+4\right)^2}+\frac{C}{x-1}+\frac{D}{x+4}$

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

$\frac{1}{\left(x-1\right)^2\left(x+4\right)^2}=\frac{A}{\left(x-1\right)^2}+\frac{B}{\left(x+4\right)^2}+\frac{C}{x-1}+\frac{D}{x+4}$

Learn how to solve integrals by partial fraction expansion 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. Find the values for the unknown coefficients: A, B, C, D. The first step is to multiply both sides of the equation from the previous step by \left(x-1\right)^2\left(x+4\right)^2. Multiplying polynomials. Simplifying.

##  Final answer to the problem

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

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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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

###  Main Topic: Integrals by Partial Fraction Expansion

The partial fraction decomposition or partial fraction expansion of a rational function is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.