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Identical expressions
one /((3x^ two)- five)
1 divide by ((3x squared ) minus 5)
one divide by ((3x to the power of two) minus five)
1/((3x2)-5)
1/3x2-5
1/((3x²)-5)
1/((3x to the power of 2)-5)
1/3x^2-5
1 divide by ((3x^2)-5)
1/((3x^2)-5)dx
Similar expressions
1/((3x^2)+5)

Solving integrals/ 1/((3x^2)-5)

Integral of 1/((3x^2)-5) dx

The solution

You have entered [src]

  1              
  /              
 |               
 |       1       
 |  1*-------- dx
 |       2       
 |    3*x  - 5   
 |               
/                
0

\int\limits_{0}^{1} 1 \cdot \frac{1}{3 x^{2} - 5}\, dx

Integral(1/(3*x^2 - 1*5), (x, 0, 1))

Detail solution

Rewrite the integrand:

$1 \cdot \frac{1}{3 x^{2} - 5} = \frac{\sqrt{15}}{30} \left(- \frac{1}{x + \frac{\sqrt{15}}{3}} + \frac{1}{x - \frac{\sqrt{15}}{3}}\right)$
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\sqrt{15}}{30} \left(- \frac{1}{x + \frac{\sqrt{15}}{3}} + \frac{1}{x - \frac{\sqrt{15}}{3}}\right)\, dx = \frac{\sqrt{15} \int \left(- \frac{1}{x + \frac{\sqrt{15}}{3}} + \frac{1}{x - \frac{\sqrt{15}}{3}}\right)\, dx}{30}$
1. Integrate term-by-term:
  1. The integral of $\frac{1}{x - \frac{\sqrt{15}}{3}}$ is $\log{\left(x - \frac{\sqrt{15}}{3} \right)}$ .
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x + \frac{\sqrt{15}}{3}}\right)\, dx = - \int \frac{1}{x + \frac{\sqrt{15}}{3}}\, dx$
    1. The integral of $\frac{1}{x + \frac{\sqrt{15}}{3}}$ is $\log{\left(x + \frac{\sqrt{15}}{3} \right)}$ .
    So, the result is: $- \log{\left(x + \frac{\sqrt{15}}{3} \right)}$
  The result is: $\log{\left(x - \frac{\sqrt{15}}{3} \right)} - \log{\left(x + \frac{\sqrt{15}}{3} \right)}$
So, the result is: $\frac{\sqrt{15} \left(\log{\left(x - \frac{\sqrt{15}}{3} \right)} - \log{\left(x + \frac{\sqrt{15}}{3} \right)}\right)}{30}$
Add the constant of integration:

$\frac{\sqrt{15} \left(\log{\left(x - \frac{\sqrt{15}}{3} \right)} - \log{\left(x + \frac{\sqrt{15}}{3} \right)}\right)}{30}+ \mathrm{constant}$

The answer is:

\frac{\sqrt{15} \left(\log{\left(x - \frac{\sqrt{15}}{3} \right)} - \log{\left(x + \frac{\sqrt{15}}{3} \right)}\right)}{30}+ \mathrm{constant}

The answer (Indefinite) [src]

                              /     /      ____\      /      ____\\
  /                      ____ |     |    \/ 15 |      |    \/ 15 ||
 |                     \/ 15 *|- log|x + ------| + log|x - ------||
 |      1                     \     \      3   /      \      3   //
 | 1*-------- dx = C + --------------------------------------------
 |      2                                   30                     
 |   3*x  - 5                                                      
 |                                                                 
/

{{\log \left({{6\,x-2\,\sqrt{15}}\over{6\,x+2\,\sqrt{15}}}\right) }\over{2\,\sqrt{15}}}

Simplify

The graph

Plot the graph f(x)

The answer [src]

         /          /  ____\\             /      ____\          /          /       ____\\             /  ____\
    ____ |          |\/ 15 ||     ____    |    \/ 15 |     ____ |          |     \/ 15 ||     ____    |\/ 15 |
  \/ 15 *|pi*I + log|------||   \/ 15 *log|1 + ------|   \/ 15 *|pi*I + log|-1 + ------||   \/ 15 *log|------|
         \          \  3   //             \      3   /          \          \       3   //             \  3   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               30                         30                            30                          30

{{\log \left(4-\sqrt{15}\right)}\over{2\,\sqrt{15}}}

         /          /  ____\\             /      ____\          /          /       ____\\             /  ____\
    ____ |          |\/ 15 ||     ____    |    \/ 15 |     ____ |          |     \/ 15 ||     ____    |\/ 15 |
  \/ 15 *|pi*I + log|------||   \/ 15 *log|1 + ------|   \/ 15 *|pi*I + log|-1 + ------||   \/ 15 *log|------|
         \          \  3   //             \      3   /          \          \       3   //             \  3   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               30                         30                            30                          30

- \frac{\sqrt{15} \log{\left(1 + \frac{\sqrt{15}}{3} \right)}}{30} + \frac{\sqrt{15} \log{\left(\frac{\sqrt{15}}{3} \right)}}{30} - \frac{\sqrt{15} \left(\log{\left(\frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30} + \frac{\sqrt{15} \left(\log{\left(-1 + \frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30}

Simplify

Numerical answer [src]

-0.266388580125985

-0.266388580125985

The graph

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

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Integral of 1/((3x^2)-5) dx

Limits of integration:

The graph:

Piecewise:

The solution