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# Find the integral $\int\frac{x^3}{x^3+1}dx$

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##  Final answer to the problem

$x-\frac{2}{9}\arctan\left(\frac{-1+2x}{\sqrt{3}}\right)-\frac{1}{3}\ln\left|\frac{\sqrt{3}}{2\sqrt{\left(x-\frac{1}{2}\right)^2+\frac{3}{4}}}\right|-\frac{1}{3}\ln\left|x+1\right|+C_0$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Integrate by partial fractions
• Integrate by substitution
• Integrate by parts
• Integrate using tabular integration
• Integrate by trigonometric substitution
• Weierstrass Substitution
• Integrate using trigonometric identities
• Integrate using basic integrals
• Product of Binomials with Common Term
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1

Divide $x^3$ by $x^3+1$

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

Learn how to solve problems step by step online.

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

Learn how to solve problems step by step online. Find the integral int((x^3)/(x^3+1))dx. Divide x^3 by x^3+1. Resulting polynomial. Expand the integral \int\left(1+\frac{-1}{x^3+1}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int1dx results in: x.

##  Final answer to the problem

$x-\frac{2}{9}\arctan\left(\frac{-1+2x}{\sqrt{3}}\right)-\frac{1}{3}\ln\left|\frac{\sqrt{3}}{2\sqrt{\left(x-\frac{1}{2}\right)^2+\frac{3}{4}}}\right|-\frac{1}{3}\ln\left|x+1\right|+C_0$

##  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

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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
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