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

Step-by-step Solution

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Final Answer

$\frac{1}{2}x^2-x+\frac{5}{4}\ln\left(x+1\right)+\frac{3}{4}\ln\left(x-1\right)+\frac{3}{2\left(x+1\right)}+C_0$
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Step-by-step Solution

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Divide $x^4+2$ by $x^3+x^2-x-1$

$\begin{array}{l}\phantom{\phantom{;}x^{3}+x^{2}-x\phantom{;}-1;}{\phantom{;}x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;}x^{3}+x^{2}-x\phantom{;}-1\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{3}+x^{2}-x\phantom{;}-1;}\underline{-x^{4}-x^{3}+x^{2}+x\phantom{;}\phantom{-;x^n}}\\\phantom{-x^{4}-x^{3}+x^{2}+x\phantom{;};}-x^{3}+x^{2}+x\phantom{;}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}+x^{2}-x\phantom{;}-1-;x^n;}\underline{\phantom{;}x^{3}+x^{2}-x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;\phantom{;}x^{3}+x^{2}-x\phantom{;}-1\phantom{;}\phantom{;}-;x^n;}\phantom{;}2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\end{array}$

Learn how to solve integrals of rational functions problems step by step online.

$\begin{array}{l}\phantom{\phantom{;}x^{3}+x^{2}-x\phantom{;}-1;}{\phantom{;}x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;}x^{3}+x^{2}-x\phantom{;}-1\overline{\smash{)}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{3}+x^{2}-x\phantom{;}-1;}\underline{-x^{4}-x^{3}+x^{2}+x\phantom{;}\phantom{-;x^n}}\\\phantom{-x^{4}-x^{3}+x^{2}+x\phantom{;};}-x^{3}+x^{2}+x\phantom{;}+2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}+x^{2}-x\phantom{;}-1-;x^n;}\underline{\phantom{;}x^{3}+x^{2}-x\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;\phantom{;}x^{3}+x^{2}-x\phantom{;}-1\phantom{;}\phantom{;}-;x^n;}\phantom{;}2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int((x^4+2)/(x^3+x^2-x+-1))dx. Divide x^4+2 by x^3+x^2-x-1. Resulting polynomial. Expand the integral \int\left(x-1+\frac{2x^{2}+1}{x^3+x^2-x-1}\right)dx into 3 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int xdx results in: \frac{1}{2}x^2.

Final Answer

$\frac{1}{2}x^2-x+\frac{5}{4}\ln\left(x+1\right)+\frac{3}{4}\ln\left(x-1\right)+\frac{3}{2\left(x+1\right)}+C_0$

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Function Plot

Plotting: $\frac{1}{2}x^2-x+\frac{5}{4}\ln\left(x+1\right)+\frac{3}{4}\ln\left(x-1\right)+\frac{3}{2\left(x+1\right)}+C_0$

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0
a
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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

How to improve your answer:

Main Topic: Integrals of Rational Functions

Integrals of rational functions of the form R(x) = P(x)/Q(x).

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