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1/(x^2-3)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1          
  /          
 |           
 |    1      
 |  ------ dx
 |   2       
 |  x  - 3   
 |           
/            
0            
$$\int\limits_{0}^{1} \frac{1}{x^{2} - 3}\, dx$$
Integral(1/(x^2 - 3), (x, 0, 1))
Detail solution

    PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=-3, context=1/(x**2 - 3), symbol=x), False), (ArccothRule(a=1, b=1, c=-3, context=1/(x**2 - 3), symbol=x), x**2 > 3), (ArctanhRule(a=1, b=1, c=-3, context=1/(x**2 - 3), symbol=x), x**2 < 3)], context=1/(x**2 - 3), symbol=x)

  1. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
                   //            /    ___\             \
                   ||   ___      |x*\/ 3 |             |
                   ||-\/ 3 *acoth|-------|             |
  /                ||            \   3   /        2    |
 |                 ||----------------------  for x  > 3|
 |   1             ||          3                       |
 | ------ dx = C + |<                                  |
 |  2              ||            /    ___\             |
 | x  - 3          ||   ___      |x*\/ 3 |             |
 |                 ||-\/ 3 *atanh|-------|             |
/                  ||            \   3   /        2    |
                   ||----------------------  for x  < 3|
                   \\          3                       /
$$\int \frac{1}{x^{2} - 3}\, dx = C + \begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)}}{3} & \text{for}\: x^{2} < 3 \end{cases}$$
The graph
The answer [src]
    ___ /          /  ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /  ___\
  \/ 3 *\pi*I + log\\/ 3 //   \/ 3 *log\1 + \/ 3 /   \/ 3 *\pi*I + log\-1 + \/ 3 //   \/ 3 *log\\/ 3 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
              6                        6                           6                         6        
$$- \frac{\sqrt{3} \log{\left(1 + \sqrt{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\sqrt{3} \right)}}{6} - \frac{\sqrt{3} \left(\log{\left(\sqrt{3} \right)} + i \pi\right)}{6} + \frac{\sqrt{3} \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right)}{6}$$
=
=
    ___ /          /  ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /  ___\
  \/ 3 *\pi*I + log\\/ 3 //   \/ 3 *log\1 + \/ 3 /   \/ 3 *\pi*I + log\-1 + \/ 3 //   \/ 3 *log\\/ 3 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
              6                        6                           6                         6        
$$- \frac{\sqrt{3} \log{\left(1 + \sqrt{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\sqrt{3} \right)}}{6} - \frac{\sqrt{3} \left(\log{\left(\sqrt{3} \right)} + i \pi\right)}{6} + \frac{\sqrt{3} \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right)}{6}$$
-sqrt(3)*(pi*i + log(sqrt(3)))/6 - sqrt(3)*log(1 + sqrt(3))/6 + sqrt(3)*(pi*i + log(-1 + sqrt(3)))/6 + sqrt(3)*log(sqrt(3))/6
Numerical answer [src]
-0.380172998150473
-0.380172998150473
The graph
Integral of 1/(x^2-3) dx

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