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Graphing y =:
x^3/(x^2-1)
Derivative of:
x^3/(x^2-1)
How do you in partial fractions?:
x^3/(x^2-1)
Identical expressions
x^ three /(x^ two - one)
x cubed divide by (x squared minus 1)
x to the power of three divide by (x to the power of two minus one)
x3/(x2-1)
x3/x2-1
x³/(x²-1)
x to the power of 3/(x to the power of 2-1)
x^3/x^2-1
x^3 divide by (x^2-1)
x^3/(x^2-1)dx
Similar expressions
x^3/(x^2+1)

Solving integrals/ x^3/(x^2-1)

Integral of x^3/(x^2-1) dx

The solution

You have entered [src]

  1          
  /          
 |           
 |     3     
 |    x      
 |  ------ dx
 |   2       
 |  x  - 1   
 |           
/            
0

$$\int\limits_{0}^{1} \frac{x^{3}}{x^{2} - 1}\, dx$$

Integral(x^3/(x^2 - 1), (x, 0, 1))

Detail solution

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

$\frac{x^{2}}{2} + \frac{\log{\left(x^{2} - 1 \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{x^{2}}{2} + \frac{\log{\left(x^{2} - 1 \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                  
 |    3             2      /      2\
 |   x             x    log\-1 + x /
 | ------ dx = C + -- + ------------
 |  2              2         2      
 | x  - 1                           
 |                                  
/

$$\int \frac{x^{3}}{x^{2} - 1}\, dx = C + \frac{x^{2}}{2} + \frac{\log{\left(x^{2} - 1 \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      pi*I
-oo - ----
       2

$$-\infty - \frac{i \pi}{2}$$

      pi*I
-oo - ----
       2

$$-\infty - \frac{i \pi}{2}$$

-oo - pi*i/2

Numerical answer [src]

-21.1989048028269

-21.1989048028269

The graph

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

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Identical expressions

Similar expressions

Integral of x^3/(x^2-1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2