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

Step-by-step Solution

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Final answer to the problem

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

How should I solve this problem?

  • 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
  • FOIL Method
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1

Rewrite the expression $\frac{1}{-x+x^3}$ inside the integral in factored form

$\int\frac{1}{x\left(1+x\right)\left(-1+x\right)}dx$

Learn how to solve differential equations problems step by step online.

$\int\frac{1}{x\left(1+x\right)\left(-1+x\right)}dx$

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Unlock the first 3 steps of this solution

Learn how to solve differential equations problems step by step online. Find the integral int(1/(-x+x^3))dx. Rewrite the expression \frac{1}{-x+x^3} inside the integral in factored form. Rewrite the fraction \frac{1}{x\left(1+x\right)\left(-1+x\right)} in 3 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{-1}{x}+\frac{1}{2\left(1+x\right)}+\frac{1}{2\left(-1+x\right)}\right)dx into 3 integrals using the sum rule for integrals, to then solve each integral separately. We can solve the integral \int\frac{1}{2\left(1+x\right)}dx by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it u), which when substituted makes the integral easier. We see that 1+x it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part.

Final answer to the problem

$-\ln\left|x\right|+\frac{1}{2}\ln\left|1+x\right|+\frac{1}{2}\ln\left|x-1\right|+C_0$

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Function Plot

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

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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: Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives.

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