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

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

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The solution

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∫011⋅1x2−2 dx\int\limits_{0}^{1} 1 \cdot \frac{1}{x^{2} - 2}\, dx 
Integral(1/(x^2 - 1*2), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    1⋅1x2−2=24(−1x+2+1x−2)1 \cdot \frac{1}{x^{2} - 2} = \frac{\sqrt{2}}{4} \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫24(−1x+2+1x−2) dx=2∫(−1x+2+1x−2) dx4\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 1x−2\frac{1}{x - \sqrt{2}} is log⁡(x−2)\log{\left(x - \sqrt{2} \right)}.

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−1x+2) dx=−∫1x+2 dx\int \left(- \frac{1}{x + \sqrt{2}}\right)\, dx = - \int \frac{1}{x + \sqrt{2}}\, dx

        1. The integral of 1x+2\frac{1}{x + \sqrt{2}} is log⁡(x+2)\log{\left(x + \sqrt{2} \right)}.

        So, the result is: −log⁡(x+2)- \log{\left(x + \sqrt{2} \right)}

      The result is: log⁡(x−2)−log⁡(x+2)\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}

    So, the result is: 2(log⁡(x−2)−log⁡(x+2))4\frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4}

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                                           
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 |     1             \/ 2 *\- log\x + \/ 2 / + log\x - \/ 2 //
 | 1*------ dx = C + -----------------------------------------
 |    2                                  4                    
 |   x  - 2                                                   
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log⁡(2 x−2322 x+232)232{{\log \left({{2\,x-2^{{{3}\over{2}}}}\over{2\,x+2^{{{3}\over{2}}} }}\right)}\over{2^{{{3}\over{2}}}}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-1.50.0
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⁡(3−232)232{{\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
−2log⁡(1+2)4+2log⁡(2)4−2(log⁡(2)+iπ)4+2(log⁡(−1+2)+iπ)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
Integral of 1/(x^2-2) dx

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

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