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

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

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The solution

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  1          
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∫012x2−1 dx\int\limits_{0}^{1} \frac{2}{x^{2} - 1}\, dx 
Integral(2/(x^2 - 1), (x, 0, 1))
Detail solution

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

    ∫2x2−1 dx=2∫1x2−1 dx\int \frac{2}{x^{2} - 1}\, dx = 2 \int \frac{1}{x^{2} - 1}\, dx

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

    So, the result is: 2({−acoth⁡(x)for x2>1−atanh⁡(x)for x2<1)2 \left(\begin{cases} - \operatorname{acoth}{\left(x \right)} & \text{for}\: x^{2} > 1 \\- \operatorname{atanh}{\left(x \right)} & \text{for}\: x^{2} < 1 \end{cases}\right)

  2. Now simplify:

    {−2acoth⁡(x)for x2>1−2atanh⁡(x)for x2<1\begin{cases} - 2 \operatorname{acoth}{\left(x \right)} & \text{for}\: x^{2} > 1 \\- 2 \operatorname{atanh}{\left(x \right)} & \text{for}\: x^{2} < 1 \end{cases}

  3. Add the constant of integration:

    {−2acoth⁡(x)for x2>1−2atanh⁡(x)for x2<1+constant\begin{cases} - 2 \operatorname{acoth}{\left(x \right)} & \text{for}\: x^{2} > 1 \\- 2 \operatorname{atanh}{\left(x \right)} & \text{for}\: x^{2} < 1 \end{cases}+ \mathrm{constant}

The answer is:

{−2acoth⁡(x)for x2>1−2atanh⁡(x)for x2<1+constant\begin{cases} - 2 \operatorname{acoth}{\left(x \right)} & \text{for}\: x^{2} > 1 \\- 2 \operatorname{atanh}{\left(x \right)} & \text{for}\: x^{2} < 1 \end{cases}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                          
 |                   //                2    \
 |   2               ||-acoth(x)  for x  > 1|
 | ------ dx = C + 2*|<                     |
 |  2                ||                2    |
 | x  - 1            \\-atanh(x)  for x  < 1/
 |                                           
/
∫2x2−1 dx=C+2({−acoth⁡(x)for x2>1−atanh⁡(x)for x2<1)\int \frac{2}{x^{2} - 1}\, dx = C + 2 \left(\begin{cases} - \operatorname{acoth}{\left(x \right)} & \text{for}\: x^{2} > 1 \\- \operatorname{atanh}{\left(x \right)} & \text{for}\: x^{2} < 1 \end{cases}\right)
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-100005000
Plot the graph f(x)
The answer [src] 
-oo - pi*I
−∞−iπ-\infty - i \pi 
=
= 
-oo - pi*I
−∞−iπ-\infty - i \pi 
-oo - pi*i
Numerical answer [src] 
-44.7841039667738
-44.7841039667738
The graph
Integral of 2/(x^2-1) dx

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

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