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Identical expressions
dx/(five *x^ two - six)
dx divide by (5 multiply by x squared minus 6)
dx divide by (five multiply by x to the power of two minus six)
dx/(5*x2-6)
dx/5*x2-6
dx/(5*x²-6)
dx/(5*x to the power of 2-6)
dx/(5x^2-6)
dx/(5x2-6)
dx/5x2-6
dx/5x^2-6
dx divide by (5*x^2-6)
Similar expressions
dx/(5*x^2+6)

Solving integrals/ dx/(5*x^2-6)

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

The solution

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  1              
  /              
 |               
 |       1       
 |  1*-------- dx
 |       2       
 |    5*x  - 6   
 |               
/                
0

$$\int\limits_{0}^{1} 1 \cdot \frac{1}{5 x^{2} - 6}\, dx$$

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

Detail solution

Rewrite the integrand:

$1 \cdot \frac{1}{5 x^{2} - 6} = \frac{\sqrt{30}}{60} \left(- \frac{1}{x + \frac{\sqrt{30}}{5}} + \frac{1}{x - \frac{\sqrt{30}}{5}}\right)$
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\sqrt{30}}{60} \left(- \frac{1}{x + \frac{\sqrt{30}}{5}} + \frac{1}{x - \frac{\sqrt{30}}{5}}\right)\, dx = \frac{\sqrt{30} \int \left(- \frac{1}{x + \frac{\sqrt{30}}{5}} + \frac{1}{x - \frac{\sqrt{30}}{5}}\right)\, dx}{60}$
1. Integrate term-by-term:
  1. The integral of $\frac{1}{x - \frac{\sqrt{30}}{5}}$ is $\log{\left(x - \frac{\sqrt{30}}{5} \right)}$ .
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x + \frac{\sqrt{30}}{5}}\right)\, dx = - \int \frac{1}{x + \frac{\sqrt{30}}{5}}\, dx$
    1. The integral of $\frac{1}{x + \frac{\sqrt{30}}{5}}$ is $\log{\left(x + \frac{\sqrt{30}}{5} \right)}$ .
    So, the result is: $- \log{\left(x + \frac{\sqrt{30}}{5} \right)}$
  The result is: $\log{\left(x - \frac{\sqrt{30}}{5} \right)} - \log{\left(x + \frac{\sqrt{30}}{5} \right)}$
So, the result is: $\frac{\sqrt{30} \left(\log{\left(x - \frac{\sqrt{30}}{5} \right)} - \log{\left(x + \frac{\sqrt{30}}{5} \right)}\right)}{60}$
Add the constant of integration:

$\frac{\sqrt{30} \left(\log{\left(x - \frac{\sqrt{30}}{5} \right)} - \log{\left(x + \frac{\sqrt{30}}{5} \right)}\right)}{60}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{30} \left(\log{\left(x - \frac{\sqrt{30}}{5} \right)} - \log{\left(x + \frac{\sqrt{30}}{5} \right)}\right)}{60}+ \mathrm{constant}$

The answer (Indefinite) [src]

                              /     /      ____\      /      ____\\
  /                      ____ |     |    \/ 30 |      |    \/ 30 ||
 |                     \/ 30 *|- log|x + ------| + log|x - ------||
 |      1                     \     \      5   /      \      5   //
 | 1*-------- dx = C + --------------------------------------------
 |      2                                   60                     
 |   5*x  - 6                                                      
 |                                                                 
/

$$\int 1 \cdot \frac{1}{5 x^{2} - 6}\, dx = C + \frac{\sqrt{30} \left(\log{\left(x - \frac{\sqrt{30}}{5} \right)} - \log{\left(x + \frac{\sqrt{30}}{5} \right)}\right)}{60}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         /          /  ____\\             /      ____\          /          /       ____\\             /  ____\
    ____ |          |\/ 30 ||     ____    |    \/ 30 |     ____ |          |     \/ 30 ||     ____    |\/ 30 |
  \/ 30 *|pi*I + log|------||   \/ 30 *log|1 + ------|   \/ 30 *|pi*I + log|-1 + ------||   \/ 30 *log|------|
         \          \  5   //             \      5   /          \          \       5   //             \  5   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               60                         60                            60                          60

$$- \frac{\sqrt{30} \log{\left(1 + \frac{\sqrt{30}}{5} \right)}}{60} + \frac{\sqrt{30} \log{\left(\frac{\sqrt{30}}{5} \right)}}{60} - \frac{\sqrt{30} \left(\log{\left(\frac{\sqrt{30}}{5} \right)} + i \pi\right)}{60} + \frac{\sqrt{30} \left(\log{\left(-1 + \frac{\sqrt{30}}{5} \right)} + i \pi\right)}{60}$$

         /          /  ____\\             /      ____\          /          /       ____\\             /  ____\
    ____ |          |\/ 30 ||     ____    |    \/ 30 |     ____ |          |     \/ 30 ||     ____    |\/ 30 |
  \/ 30 *|pi*I + log|------||   \/ 30 *log|1 + ------|   \/ 30 *|pi*I + log|-1 + ------||   \/ 30 *log|------|
         \          \  5   //             \      5   /          \          \       5   //             \  5   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               60                         60                            60                          60

Упростить

Numerical answer [src]

-0.281983082722996

-0.281983082722996

The graph

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

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

Limits of integration:

The graph:

Piecewise:

The solution