Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x^4lnx
Integral of dx/(1-4*x^2)
Integral of (4x-x^2)dx
Integral of log(x+1)/(x^2+1)
How do you in partial fractions?:
1/(x^4-1)
Graphing y =:
1/(x^4-1)
Identical expressions
one /(x^ four - one)
1 divide by (x to the power of 4 minus 1)
one divide by (x to the power of four minus one)
1/(x4-1)
1/x4-1
1/(x⁴-1)
1/x^4-1
1 divide by (x^4-1)
1/(x^4-1)dx
Similar expressions
1/(x^4+1)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$\frac{1}{x^{4} - 1} = - \frac{1}{2 \left(x^{2} + 1\right)} - \frac{1}{4 \left(x + 1\right)} + \frac{1}{4 \left(x - 1\right)}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{2 \left(x^{2} + 1\right)}\right)\, dx = - \frac{\int \frac{1}{x^{2} + 1}\, dx}{2}$
  So, the result is: $- \frac{\operatorname{atan}{\left(x \right)}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{4 \left(x + 1\right)}\right)\, dx = - \frac{\int \frac{1}{x + 1}\, dx}{4}$
  1. Let $u = x + 1$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. 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 $du$ :
    
    $\int \frac{1}{u}\, du$
    1. 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: $\frac{\log{\left(x - 1 \right)}}{4} - \frac{\log{\left(x + 1 \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2}$
Add the constant of integration:

$\frac{\log{\left(x - 1 \right)}}{4} - \frac{\log{\left(x + 1 \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(x - 1 \right)}}{4} - \frac{\log{\left(x + 1 \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                  
 |                                                   
 |   1             atan(x)   log(1 + x)   log(-1 + x)
 | ------ dx = C - ------- - ---------- + -----------
 |  4                 2          4             4     
 | x  - 1                                            
 |                                                   
/

$$\int \frac{1}{x^{4} - 1}\, dx = C + \frac{\log{\left(x - 1 \right)}}{4} - \frac{\log{\left(x + 1 \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      pi*I
-oo - ----
       4

$$-\infty - \frac{i \pi}{4}$$

      pi*I
-oo - ----
       4

$$-\infty - \frac{i \pi}{4}$$

-oo - pi*i/4

Numerical answer [src]

-11.5887250733929

-11.5887250733929

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 1/(x^4-1) dx

Limits of integration:

The graph:

Piecewise:

The solution