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1/(x^2-2)
Identical expressions
one /(x^ two - two)
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1/(x2-2)
1/x2-2
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1/(x^2-2)dx
Similar expressions
1/(x^2+2)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$1 \cdot \frac{1}{x^{2} - 2} = \frac{\sqrt{2}}{4} \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)$
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\sqrt{2}}{4} \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)\, dx = \frac{\sqrt{2} \int \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)\, dx}{4}$
1. Integrate term-by-term:
  1. The integral of $\frac{1}{x - \sqrt{2}}$ is $\log{\left(x - \sqrt{2} \right)}$ .
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x + \sqrt{2}}\right)\, dx = - \int \frac{1}{x + \sqrt{2}}\, dx$
    1. The integral of $\frac{1}{x + \sqrt{2}}$ is $\log{\left(x + \sqrt{2} \right)}$ .
    So, the result is: $- \log{\left(x + \sqrt{2} \right)}$
  The result is: $\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}$
So, the result is: $\frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                           
 |                     ___ /     /      ___\      /      ___\\
 |     1             \/ 2 *\- log\x + \/ 2 / + log\x - \/ 2 //
 | 1*------ dx = C + -----------------------------------------
 |    2                                  4                    
 |   x  - 2                                                   
 |                                                            
/

$${{\log \left({{2\,x-2^{{{3}\over{2}}}}\over{2\,x+2^{{{3}\over{2}}} }}\right)}\over{2^{{{3}\over{2}}}}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___ /          /  ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /  ___\
  \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /   \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
              4                        4                           4                         4

$${{\log \left(3-2^{{{3}\over{2}}}\right)}\over{2^{{{3}\over{2}}}}}$$

    ___ /          /  ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /  ___\
  \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /   \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
              4                        4                           4                         4

$$- \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{4} - \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{4}$$

Simplify

Numerical answer [src]

-0.623225240140231

-0.623225240140231

The graph

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

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

Limits of integration:

The graph:

Piecewise:

The solution