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  • Similar expressions

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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  y                    
  /                    
 |                     
 |         1           
 |  ---------------- dx
 |           _______   
 |          / 1        
 |  pi*x*  /  - - 1    
 |       \/   x        
 |                     
/                      
0
$$\int\limits_{0}^{y} \frac{1}{\pi x \sqrt{-1 + \frac{1}{x}}}\, dx$$ 
Integral(1/((pi*x)*sqrt(1/x - 1)), (x, 0, y))
The answer (Indefinite) [src] 
                               /                 
                              |                  
                              |       1          
                              | -------------- dx
                              |       ________   
                              |      /      1    
                              | x*  /  -1 + -    
  /                           |   \/        x    
 |                            |                  
 |        1                  /                   
 | ---------------- dx = C + --------------------
 |          _______                   pi         
 |         / 1                                   
 | pi*x*  /  - - 1                               
 |      \/   x                                   
 |                                               
/
$$\int \frac{1}{\pi x \sqrt{-1 + \frac{1}{x}}}\, dx = C + \frac{\int \frac{1}{x \sqrt{-1 + \frac{1}{x}}}\, dx}{\pi}$$
Simplify
The answer [src] 
/      /  ____________\            /    ____\            /  ____________\           
|2*asin\\/ Max(-1, y) /   2*I*acosh\I*\/ -y /   2*I*acosh\\/ Max(-1, y) /           
|---------------------- - ------------------- + -------------------------  for y < 0
|          pi                      pi                       pi                      
<                                                                                   
|        /  ___________\            /  ___\            /  ___________\              
|  2*asin\\/ Min(1, y) /   2*I*acosh\\/ y /   2*I*acosh\\/ Min(1, y) /              
|  --------------------- - ---------------- + ------------------------     otherwise
\            pi                   pi                     pi
$$\begin{cases} - \frac{2 i \operatorname{acosh}{\left(i \sqrt{- y} \right)}}{\pi} + \frac{2 i \operatorname{acosh}{\left(\sqrt{\max\left(-1, y\right)} \right)}}{\pi} + \frac{2 \operatorname{asin}{\left(\sqrt{\max\left(-1, y\right)} \right)}}{\pi} & \text{for}\: y < 0 \\- \frac{2 i \operatorname{acosh}{\left(\sqrt{y} \right)}}{\pi} + \frac{2 i \operatorname{acosh}{\left(\sqrt{\min\left(1, y\right)} \right)}}{\pi} + \frac{2 \operatorname{asin}{\left(\sqrt{\min\left(1, y\right)} \right)}}{\pi} & \text{otherwise} \end{cases}$$ 
=
= 
/      /  ____________\            /    ____\            /  ____________\           
|2*asin\\/ Max(-1, y) /   2*I*acosh\I*\/ -y /   2*I*acosh\\/ Max(-1, y) /           
|---------------------- - ------------------- + -------------------------  for y < 0
|          pi                      pi                       pi                      
<                                                                                   
|        /  ___________\            /  ___\            /  ___________\              
|  2*asin\\/ Min(1, y) /   2*I*acosh\\/ y /   2*I*acosh\\/ Min(1, y) /              
|  --------------------- - ---------------- + ------------------------     otherwise
\            pi                   pi                     pi
$$\begin{cases} - \frac{2 i \operatorname{acosh}{\left(i \sqrt{- y} \right)}}{\pi} + \frac{2 i \operatorname{acosh}{\left(\sqrt{\max\left(-1, y\right)} \right)}}{\pi} + \frac{2 \operatorname{asin}{\left(\sqrt{\max\left(-1, y\right)} \right)}}{\pi} & \text{for}\: y < 0 \\- \frac{2 i \operatorname{acosh}{\left(\sqrt{y} \right)}}{\pi} + \frac{2 i \operatorname{acosh}{\left(\sqrt{\min\left(1, y\right)} \right)}}{\pi} + \frac{2 \operatorname{asin}{\left(\sqrt{\min\left(1, y\right)} \right)}}{\pi} & \text{otherwise} \end{cases}$$ 
Piecewise((2*asin(sqrt(Max(-1, y)))/pi - 2*i*acosh(i*sqrt(-y))/pi + 2*i*acosh(sqrt(Max(-1, y)))/pi, y < 0), (2*asin(sqrt(Min(1, y)))/pi - 2*i*acosh(sqrt(y))/pi + 2*i*acosh(sqrt(Min(1, y)))/pi, True))

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

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