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Find the integral of $\frac{\frac{x^3-1}{x^3-2x^2-3x}\left(x+1\right)}{x^2+x-2}+\frac{x^2+x+1}{6x+x^2-x^3}$

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Solution

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Final answer to the problem

$\frac{13}{15}\ln\left(x-3\right)-\frac{13}{15}\ln\left(-x+3\right)+C_0$
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Step-by-step Solution

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Find the integral

$\int\left(\frac{\frac{x^3-1}{x^3-2x^2-3x}\left(x+1\right)}{x^2+x-2}+\frac{x^2+x+1}{6x+x^2-x^3}\right)dx$

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$\int\left(\frac{\frac{x^3-1}{x^3-2x^2-3x}\left(x+1\right)}{x^2+x-2}+\frac{x^2+x+1}{6x+x^2-x^3}\right)dx$

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Learn how to solve integral calculus problems step by step online. Find the integral of ((x^3-1)/(x^3-2x^2-3x)(x+1))/(x^2+x+-2)+(x^2+x+1)/(6x+x^2-x^3). Find the integral. Simplify the expression inside the integral. The integral \int\frac{\left(x^3-1\right)\left(x+1\right)}{\left(x^3-2x^2-3x\right)\left(x^2+x-2\right)}dx results in: \int\frac{x^3-1}{x\left(x-3\right)\left(x-1\right)\left(x+2\right)}dx. The integral \int\frac{x^2+x+1}{6x+x^2-x^3}dx results in: \frac{1}{6}\ln\left(x\right)-\frac{13}{15}\ln\left(-x+3\right)-\frac{3}{10}\ln\left(x+2\right).

Final answer to the problem

$\frac{13}{15}\ln\left(x-3\right)-\frac{13}{15}\ln\left(-x+3\right)+C_0$

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

Solve integral of ((x^3-1)/(x^3+-2x^2)(x+1)/(x^2+x)+(x^2+x)/(6x+x^2))dx using partial fractionsSolve integral of ((x^3-1)/(x^3+-2x^2)(x+1)/(x^2+x)+(x^2+x)/(6x+x^2))dx using basic integralsSolve integral of ((x^3-1)/(x^3+-2x^2)(x+1)/(x^2+x)+(x^2+x)/(6x+x^2))dx using u-substitutionSolve integral of ((x^3-1)/(x^3+-2x^2)(x+1)/(x^2+x)+(x^2+x)/(6x+x^2))dx using integration by partsSolve integral of ((x^3-1)/(x^3+-2x^2)(x+1)/(x^2+x)+(x^2+x)/(6x+x^2))dx using trigonometric substitution

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Plotting: $\frac{13}{15}\ln\left(x-3\right)-\frac{13}{15}\ln\left(-x+3\right)+C_0$

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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
D□x
∫
∫◻
|◻|
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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Main Topic: Integral Calculus

Integration assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

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