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1/(x(x+2)^(1/2))dx
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1/(x(x-2)^(1/2))

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

The solution

You have entered [src]

 -1               
  /               
 |                
 |       1        
 |  ----------- dx
 |      _______   
 |  x*\/ x + 2    
 |                
/                 
-2

\int\limits_{-2}^{-1} \frac{1}{x \sqrt{x + 2}}\, dx

Integral(1/(x*sqrt(x + 2)), (x, -2, -1))

The answer (Indefinite) [src]

                        //            /  ___   _______\                 \
  /                     ||   ___      |\/ 2 *\/ 2 + x |      |2 + x|    |
 |                      ||-\/ 2 *acoth|---------------|  for ------- > 1|
 |      1               ||            \       2       /         2       |
 | ----------- dx = C + |<                                              |
 |     _______          ||            /  ___   _______\                 |
 | x*\/ x + 2           ||   ___      |\/ 2 *\/ 2 + x |                 |
 |                      ||-\/ 2 *atanh|---------------|     otherwise   |
/                       \\            \       2       /                 /

\int \frac{1}{x \sqrt{x + 2}}\, dx = C + \begin{cases} - \sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} \sqrt{x + 2}}{2} \right)} & \text{for}\: \frac{\left|{x + 2}\right|}{2} > 1 \\- \sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} \sqrt{x + 2}}{2} \right)} & \text{otherwise} \end{cases}

Simplify

The graph

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The answer [src]

            /  ___\
   ___      |\/ 2 |
-\/ 2 *atanh|-----|
            \  2  /

- \sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}

            /  ___\
   ___      |\/ 2 |
-\/ 2 *atanh|-----|
            \  2  /

- \sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2}}{2} \right)}

-sqrt(2)*atanh(sqrt(2)/2)

Numerical answer [src]

-1.24645048001522

-1.24645048001522

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

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

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The solution