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

Step-by-step Solution

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Final Answer

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

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

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

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

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

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int((x^4-2x^24x+1)/(x^3-x^2-x+1))dx. Divide x^4-2x^2+4x+1 by x^3-x^2-x+1. Resulting polynomial. Simplify the expression inside the integral. The integral \int xdx results in: \frac{1}{2}x^2.

Final Answer

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

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Function Plot

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

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5
6
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

How to improve your answer:

Main Topic: Integrals of Rational Functions

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

Used Formulas

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