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Integral of (2x^2-5)/(x^4-5x^2+6)
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How do you in partial fractions?:
(2x^2-5)/(x^4-5x^2+6)
Identical expressions
(two x^ two - five)/(x^ four -5x^2+ six)
(2x squared minus 5) divide by (x to the power of 4 minus 5x squared plus 6)
(two x to the power of two minus five) divide by (x to the power of four minus 5x squared plus six)
(2x2-5)/(x4-5x2+6)
2x2-5/x4-5x2+6
(2x²-5)/(x⁴-5x²+6)
(2x to the power of 2-5)/(x to the power of 4-5x to the power of 2+6)
2x^2-5/x^4-5x^2+6
(2x^2-5) divide by (x^4-5x^2+6)
(2x^2-5)/(x^4-5x^2+6)dx
Similar expressions
(2x^2+5)/(x^4-5x^2+6)
(2x^2-5)/(x^4+5x^2+6)
(2x^2-5)/(x^4-5x^2-6)

Solving integrals/ (2x^2-5)/(x^4-5x^2+6)

Integral of (2x^2-5)/(x^4-5x^2+6) dx

The solution

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  0                 
  /                 
 |                  
 |        2         
 |     2*x  - 5     
 |  ------------- dx
 |   4      2       
 |  x  - 5*x  + 6   
 |                  
/                   
0

$$\int\limits_{0}^{0} \frac{2 x^{2} - 5}{x^{4} - 5 x^{2} + 6}\, dx$$

Integral((2*x^2 - 1*5)/(x^4 - 5*x^2 + 6), (x, 0, 0))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{2 x^{2} - 5}{x^{4} - 5 x^{2} + 6} = \frac{1}{x^{2} - 2} + \frac{1}{x^{2} - 3}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{1}{x^{2} - 2} = \frac{\sqrt{2}}{4} \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sqrt{2}}{4} \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)\, dx = \frac{\sqrt{2} \int \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)\, dx}{4}$
    1. Integrate term-by-term:
      
      The integral of $\frac{1}{x - \sqrt{2}}$ is $\log{\left(x - \sqrt{2} \right)}$ .
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + \sqrt{2}}\right)\, dx = - \int \frac{1}{x + \sqrt{2}}\, dx$
      
      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: $\frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4}$
  1. Rewrite the integrand:
    
    $\frac{1}{x^{2} - 3} = \frac{\sqrt{3}}{6} \left(- \frac{1}{x + \sqrt{3}} + \frac{1}{x - \sqrt{3}}\right)$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sqrt{3}}{6} \left(- \frac{1}{x + \sqrt{3}} + \frac{1}{x - \sqrt{3}}\right)\, dx = \frac{\sqrt{3} \int \left(- \frac{1}{x + \sqrt{3}} + \frac{1}{x - \sqrt{3}}\right)\, dx}{6}$
    1. Integrate term-by-term:
      
      The integral of $\frac{1}{x - \sqrt{3}}$ is $\log{\left(x - \sqrt{3} \right)}$ .
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + \sqrt{3}}\right)\, dx = - \int \frac{1}{x + \sqrt{3}}\, dx$
      
      The integral of $\frac{1}{x + \sqrt{3}}$ is $\log{\left(x + \sqrt{3} \right)}$ .
      
      So, the result is: $- \log{\left(x + \sqrt{3} \right)}$
      
      The result is: $\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}$
    So, the result is: $\frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{6}$
  The result is: $\frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4} + \frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{6}$
Method #2
1. Rewrite the integrand:
  
  $\frac{2 x^{2} - 5}{x^{4} - 5 x^{2} + 6} = \frac{2 x^{2}}{x^{4} - 5 x^{2} + 6} - \frac{5}{x^{4} - 5 x^{2} + 6}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2 x^{2}}{x^{4} - 5 x^{2} + 6}\, dx = 2 \int \frac{x^{2}}{x^{4} - 5 x^{2} + 6}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x^{2}}{x^{4} - 5 x^{2} + 6} = - \frac{2}{x^{2} - 2} + \frac{3}{x^{2} - 3}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{2}{x^{2} - 2}\right)\, dx = - 2 \int \frac{1}{x^{2} - 2}\, dx$
      
      Rewrite the integrand:
      
      $\frac{1}{x^{2} - 2} = \frac{\sqrt{2}}{4} \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sqrt{2}}{4} \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)\, dx = \frac{\sqrt{2} \int \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)\, dx}{4}$
      
      Integrate term-by-term:
      
      The integral of $\frac{1}{x - \sqrt{2}}$ is $\log{\left(x - \sqrt{2} \right)}$ .
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + \sqrt{2}}\right)\, dx = - \int \frac{1}{x + \sqrt{2}}\, dx$
      
      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: $\frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4}$
      
      So, the result is: $- \frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{3}{x^{2} - 3}\, dx = 3 \int \frac{1}{x^{2} - 3}\, dx$
      
      Rewrite the integrand:
      
      $\frac{1}{x^{2} - 3} = \frac{\sqrt{3}}{6} \left(- \frac{1}{x + \sqrt{3}} + \frac{1}{x - \sqrt{3}}\right)$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sqrt{3}}{6} \left(- \frac{1}{x + \sqrt{3}} + \frac{1}{x - \sqrt{3}}\right)\, dx = \frac{\sqrt{3} \int \left(- \frac{1}{x + \sqrt{3}} + \frac{1}{x - \sqrt{3}}\right)\, dx}{6}$
      
      Integrate term-by-term:
      
      The integral of $\frac{1}{x - \sqrt{3}}$ is $\log{\left(x - \sqrt{3} \right)}$ .
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + \sqrt{3}}\right)\, dx = - \int \frac{1}{x + \sqrt{3}}\, dx$
      
      The integral of $\frac{1}{x + \sqrt{3}}$ is $\log{\left(x + \sqrt{3} \right)}$ .
      
      So, the result is: $- \log{\left(x + \sqrt{3} \right)}$
      
      The result is: $\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}$
      
      So, the result is: $\frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{6}$
      
      So, the result is: $\frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{2}$
      
      The result is: $- \frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{2} + \frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{2}$
    So, the result is: $- \sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right) + \sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{5}{x^{4} - 5 x^{2} + 6}\right)\, dx = - 5 \int \frac{1}{x^{4} - 5 x^{2} + 6}\, dx$
    1. Rewrite the integrand:
      
      $\frac{1}{x^{4} - 5 x^{2} + 6} = - \frac{1}{x^{2} - 2} + \frac{1}{x^{2} - 3}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} - 2}\right)\, dx = - \int \frac{1}{x^{2} - 2}\, dx$
      
      Rewrite the integrand:
      
      $\frac{1}{x^{2} - 2} = \frac{\sqrt{2}}{4} \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sqrt{2}}{4} \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)\, dx = \frac{\sqrt{2} \int \left(- \frac{1}{x + \sqrt{2}} + \frac{1}{x - \sqrt{2}}\right)\, dx}{4}$
      
      Integrate term-by-term:
      
      The integral of $\frac{1}{x - \sqrt{2}}$ is $\log{\left(x - \sqrt{2} \right)}$ .
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + \sqrt{2}}\right)\, dx = - \int \frac{1}{x + \sqrt{2}}\, dx$
      
      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: $\frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4}$
      
      So, the result is: $- \frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4}$
      
      Rewrite the integrand:
      
      $\frac{1}{x^{2} - 3} = \frac{\sqrt{3}}{6} \left(- \frac{1}{x + \sqrt{3}} + \frac{1}{x - \sqrt{3}}\right)$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sqrt{3}}{6} \left(- \frac{1}{x + \sqrt{3}} + \frac{1}{x - \sqrt{3}}\right)\, dx = \frac{\sqrt{3} \int \left(- \frac{1}{x + \sqrt{3}} + \frac{1}{x - \sqrt{3}}\right)\, dx}{6}$
      
      Integrate term-by-term:
      
      The integral of $\frac{1}{x - \sqrt{3}}$ is $\log{\left(x - \sqrt{3} \right)}$ .
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + \sqrt{3}}\right)\, dx = - \int \frac{1}{x + \sqrt{3}}\, dx$
      
      The integral of $\frac{1}{x + \sqrt{3}}$ is $\log{\left(x + \sqrt{3} \right)}$ .
      
      So, the result is: $- \log{\left(x + \sqrt{3} \right)}$
      
      The result is: $\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}$
      
      So, the result is: $\frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{6}$
      
      The result is: $- \frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4} + \frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{6}$
    So, the result is: $\frac{5 \sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4} - \frac{5 \sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{6}$
  The result is: $\frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4} + \frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{6}$
Add the constant of integration:

$\frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4} + \frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{6}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{2} \left(\log{\left(x - \sqrt{2} \right)} - \log{\left(x + \sqrt{2} \right)}\right)}{4} + \frac{\sqrt{3} \left(\log{\left(x - \sqrt{3} \right)} - \log{\left(x + \sqrt{3} \right)}\right)}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                            
 |                                                                                                             
 |       2                  ___ /     /      ___\      /      ___\\     ___ /     /      ___\      /      ___\\
 |    2*x  - 5            \/ 2 *\- log\x + \/ 2 / + log\x - \/ 2 //   \/ 3 *\- log\x + \/ 3 / + log\x - \/ 3 //
 | ------------- dx = C + ----------------------------------------- + -----------------------------------------
 |  4      2                                  4                                           6                    
 | x  - 5*x  + 6                                                                                               
 |                                                                                                             
/

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

Simplify

The answer [src]

$$0$$

Упростить

Numerical answer [src]

0.0

0.0

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

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Identical expressions

Similar expressions

Integral of (2x^2-5)/(x^4-5x^2+6) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2