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Derivative of:
x^3/(x^4-1)
How do you in partial fractions?:
x^3/(x^4-1)
Graphing y =:
x^3/(x^4-1)
Identical expressions
x^ three /(x^ four - one)
x cubed divide by (x to the power of 4 minus 1)
x to the power of three divide by (x to the power of four minus one)
x3/(x4-1)
x3/x4-1
x³/(x⁴-1)
x to the power of 3/(x to the power of 4-1)
x^3/x^4-1
x^3 divide by (x^4-1)
x^3/(x^4-1)dx
Similar expressions
x^3/(x^4+1)

Solving integrals/ x^3/(x^4-1)

Integral of x^3/(x^4-1) dx

The solution

You have entered [src]

  0          
  /          
 |           
 |     3     
 |    x      
 |  ------ dx
 |   4       
 |  x  - 1   
 |           
/            
0

$$\int\limits_{0}^{0} \frac{x^{3}}{x^{4} - 1}\, dx$$

Integral(x^3/(x^4 - 1), (x, 0, 0))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = x^{4} - 1$ .
  
  Then let $du = 4 x^{3} dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{1}{4 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{4}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $\frac{\log{\left(u \right)}}{4}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(x^{4} - 1 \right)}}{4}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x^{3}}{x^{4} - 1} = \frac{x}{2 \left(x^{2} + 1\right)} + \frac{1}{4 \left(x + 1\right)} + \frac{1}{4 \left(x - 1\right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x}{2 \left(x^{2} + 1\right)}\, dx = \frac{\int \frac{x}{x^{2} + 1}\, dx}{2}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
      
      Let $u = x^{2} + 1$ .
      
      Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x^{2} + 1 \right)}$
      
      So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
    So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{4 \left(x + 1\right)}\, dx = \frac{\int \frac{1}{x + 1}\, dx}{4}$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
    So, the result is: $\frac{\log{\left(x + 1 \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{4 \left(x - 1\right)}\, dx = \frac{\int \frac{1}{x - 1}\, dx}{4}$
    1. Let $u = x - 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x - 1 \right)}$
    So, the result is: $\frac{\log{\left(x - 1 \right)}}{4}$
  The result is: $\frac{\log{\left(x - 1 \right)}}{4} + \frac{\log{\left(x + 1 \right)}}{4} + \frac{\log{\left(x^{2} + 1 \right)}}{4}$
Now simplify:

$\frac{\log{\left(x^{4} - 1 \right)}}{4}$
Add the constant of integration:

$\frac{\log{\left(x^{4} - 1 \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(x^{4} - 1 \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                           
 |                            
 |    3               / 4    \
 |   x             log\x  - 1/
 | ------ dx = C + -----------
 |  4                   4     
 | x  - 1                     
 |                            
/

$$\int \frac{x^{3}}{x^{4} - 1}\, dx = C + \frac{\log{\left(x^{4} - 1 \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

0.0

0.0

The graph

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of x^3/(x^4-1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2