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

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Final Answer

$\frac{1}{2}\ln\left(x+1\right)-\frac{1}{50}\arctan\left(x\right)-\frac{7}{100}\ln\left(x^2+1\right)+\frac{1}{5\left(x+2\right)}-\frac{9}{25}\ln\left(x+2\right)+C_0$
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Step-by-step Solution

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

$\frac{1}{\left(x+1\right)\left(x^2+1\right)\left(x+2\right)^2}=\frac{A}{x+1}+\frac{Bx+C}{x^2+1}+\frac{D}{\left(x+2\right)^2}+\frac{F}{x+2}$

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$\frac{1}{\left(x+1\right)\left(x^2+1\right)\left(x+2\right)^2}=\frac{A}{x+1}+\frac{Bx+C}{x^2+1}+\frac{D}{\left(x+2\right)^2}+\frac{F}{x+2}$

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Learn how to solve problems step by step online. Find the integral int(1/((x+1)(x^2+1)(x+2)^2))dx. Rewrite the fraction \frac{1}{\left(x+1\right)\left(x^2+1\right)\left(x+2\right)^2} in 4 simpler fractions using partial fraction decomposition. Find the values for the unknown coefficients: A, B, C, D, F. The first step is to multiply both sides of the equation from the previous step by \left(x+1\right)\left(x^2+1\right)\left(x+2\right)^2. Multiplying polynomials. Simplifying.

Final Answer

$\frac{1}{2}\ln\left(x+1\right)-\frac{1}{50}\arctan\left(x\right)-\frac{7}{100}\ln\left(x^2+1\right)+\frac{1}{5\left(x+2\right)}-\frac{9}{25}\ln\left(x+2\right)+C_0$

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Function Plot

Plotting: $\frac{1}{2}\ln\left(x+1\right)-\frac{1}{50}\arctan\left(x\right)-\frac{7}{100}\ln\left(x^2+1\right)+\frac{1}{5\left(x+2\right)}-\frac{9}{25}\ln\left(x+2\right)+C_0$

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7
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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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