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

Step-by-step Solution

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Final Answer

$\frac{1}{6}x^{3}+\frac{1}{8}x^2+\frac{1}{8}x+\frac{1}{16}\ln\left(2x-1\right)+C_0$
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Step-by-step Solution

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

$\begin{array}{l}\phantom{\phantom{;}2x\phantom{;}-1;}{\phantom{;}\frac{1}{2}x^{2}+\frac{1}{4}x\phantom{;}+\frac{1}{8}\phantom{;}\phantom{;}}\\\phantom{;}2x\phantom{;}-1\overline{\smash{)}\phantom{;}x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}2x\phantom{;}-1;}\underline{-x^{3}+\frac{1}{2}x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{3}+\frac{1}{2}x^{2};}\phantom{;}\frac{1}{2}x^{2}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n;}\underline{-\frac{1}{2}x^{2}+\frac{1}{4}x\phantom{;}\phantom{-;x^n}}\\\phantom{;-\frac{1}{2}x^{2}+\frac{1}{4}x\phantom{;}-;x^n;}\phantom{;}\frac{1}{4}x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n;}\underline{-\frac{1}{4}x\phantom{;}+\frac{1}{8}\phantom{;}\phantom{;}}\\\phantom{;;-\frac{1}{4}x\phantom{;}+\frac{1}{8}\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}\frac{1}{8}\phantom{;}\phantom{;}\\\end{array}$

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

$\begin{array}{l}\phantom{\phantom{;}2x\phantom{;}-1;}{\phantom{;}\frac{1}{2}x^{2}+\frac{1}{4}x\phantom{;}+\frac{1}{8}\phantom{;}\phantom{;}}\\\phantom{;}2x\phantom{;}-1\overline{\smash{)}\phantom{;}x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}2x\phantom{;}-1;}\underline{-x^{3}+\frac{1}{2}x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{3}+\frac{1}{2}x^{2};}\phantom{;}\frac{1}{2}x^{2}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n;}\underline{-\frac{1}{2}x^{2}+\frac{1}{4}x\phantom{;}\phantom{-;x^n}}\\\phantom{;-\frac{1}{2}x^{2}+\frac{1}{4}x\phantom{;}-;x^n;}\phantom{;}\frac{1}{4}x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n;}\underline{-\frac{1}{4}x\phantom{;}+\frac{1}{8}\phantom{;}\phantom{;}}\\\phantom{;;-\frac{1}{4}x\phantom{;}+\frac{1}{8}\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}\frac{1}{8}\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int((x^3)/(2x-1))dx. Divide x^3 by 2x-1. Resulting polynomial. Expand the integral \int\left(\frac{1}{2}x^{2}+\frac{1}{4}x+\frac{1}{8}+\frac{1}{8\left(2x-1\right)}\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{1}{2}x^{2}dx results in: \frac{1}{6}x^{3}.

Final Answer

$\frac{1}{6}x^{3}+\frac{1}{8}x^2+\frac{1}{8}x+\frac{1}{16}\ln\left(2x-1\right)+C_0$

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Function Plot

Plotting: $\frac{1}{6}x^{3}+\frac{1}{8}x^2+\frac{1}{8}x+\frac{1}{16}\ln\left(2x-1\right)+C_0$

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1
2
3
4
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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