Интеграл sqrt(r^2-x^2) (dx)

Вы ввели

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 |  \/  r  - x   dx
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$$\int\limits_{0}^{1} \sqrt{r^{2} - x^{2}}\, dx$$

Подробное решение

SqrtQuadraticRule(a=r**2, b=0, c=-1, context=sqrt(r**2 - x**2), symbol=x)

1. Добавляем постоянную интегрирования:

   $- \frac{i r^{2} \log{\left(- 2 x + 2 i \sqrt{r^{2} - x^{2}} \right)}}{2} + \frac{x \sqrt{r^{2} - x^{2}}}{2}+ \mathrm{constant}$

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Ответ:

$- \frac{i r^{2} \log{\left(- 2 x + 2 i \sqrt{r^{2} - x^{2}} \right)}}{2} + \frac{x \sqrt{r^{2} - x^{2}}}{2}+ \mathrm{constant}$

Ответ

  1                                                                                           
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 |  |          /       2                                                                      
 |  |         /       x                                                                       
 |  | I*r*   /   -1 + --                                                                      
 |  |       /          2               2                                             2        
 |  |     \/          r             I*x                      I*r                    x         
 |  | ------------------- + ------------------- - --------------------------   for ---- > 1   
 |  |          2                      _________         _______     ________       | 2|       
 |  |                                /       2         /     x     /      x        |r |       
 |  |                               /       x     2*  /  1 + - *  /  -1 + -                   
 |  |                       2*r*   /   -1 + --      \/       r  \/        r                   
 |  <                             /          2                                              dx
 |  |                           \/          r                                                 
 |  |                                                                                         
 |  |                         2                   2                  4                        
 |  |      r                 x                 3*x                  x                         
 |  |-------------- + --------------- - ------------------ - ----------------   otherwise     
 |  |      ________               3/2             ________                3/2                 
 |  |     /      2        /     2\               /      2         /     2\                    
 |  |    /      x         |    x |              /      x        3 |    x |                    
 |  |   /   1 - --    2*r*|1 - --|      2*r*   /   1 - --    2*r *|1 - --|                    
 |  |  /         2        |     2|            /         2         |     2|                    
 |  \\/         r         \    r /          \/         r          \    r /                    
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$$\int\limits_{0}^{1} \begin{cases} \frac{i r \sqrt{-1 + \frac{x^{2}}{r^{2}}}}{2} - \frac{i r}{2 \sqrt{-1 + \frac{x}{r}} \sqrt{1 + \frac{x}{r}}} + \frac{i x^{2}}{2 r \sqrt{-1 + \frac{x^{2}}{r^{2}}}} & \text{for}\: \frac{x^{2}}{\left|{r^{2}}\right|} > 1 \\\frac{r}{\sqrt{1 - \frac{x^{2}}{r^{2}}}} - \frac{3 x^{2}}{2 r \sqrt{1 - \frac{x^{2}}{r^{2}}}} + \frac{x^{2}}{2 r \left(1 - \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{x^{4}}{2 r^{3} \left(1 - \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$

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  1                                                                                           
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 |  /           _________                                                                     
 |  |          /       2                                                                      
 |  |         /       x                                                                       
 |  | I*r*   /   -1 + --                                                                      
 |  |       /          2               2                                             2        
 |  |     \/          r             I*x                      I*r                    x         
 |  | ------------------- + ------------------- - --------------------------   for ---- > 1   
 |  |          2                      _________         _______     ________       | 2|       
 |  |                                /       2         /     x     /      x        |r |       
 |  |                               /       x     2*  /  1 + - *  /  -1 + -                   
 |  |                       2*r*   /   -1 + --      \/       r  \/        r                   
 |  <                             /          2                                              dx
 |  |                           \/          r                                                 
 |  |                                                                                         
 |  |                         2                   2                  4                        
 |  |      r                 x                 3*x                  x                         
 |  |-------------- + --------------- - ------------------ - ----------------   otherwise     
 |  |      ________               3/2             ________                3/2                 
 |  |     /      2        /     2\               /      2         /     2\                    
 |  |    /      x         |    x |              /      x        3 |    x |                    
 |  |   /   1 - --    2*r*|1 - --|      2*r*   /   1 - --    2*r *|1 - --|                    
 |  |  /         2        |     2|            /         2         |     2|                    
 |  \\/         r         \    r /          \/         r          \    r /                    
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$$\int\limits_{0}^{1} \begin{cases} \frac{i r \sqrt{-1 + \frac{x^{2}}{r^{2}}}}{2} - \frac{i r}{2 \sqrt{-1 + \frac{x}{r}} \sqrt{1 + \frac{x}{r}}} + \frac{i x^{2}}{2 r \sqrt{-1 + \frac{x^{2}}{r^{2}}}} & \text{for}\: \frac{x^{2}}{\left|{r^{2}}\right|} > 1 \\\frac{r}{\sqrt{1 - \frac{x^{2}}{r^{2}}}} - \frac{3 x^{2}}{2 r \sqrt{1 - \frac{x^{2}}{r^{2}}}} + \frac{x^{2}}{2 r \left(1 - \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{x^{4}}{2 r^{3} \left(1 - \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$

Ответ (Неопределённый)

  /                                                                        
 |                            _________           /              _________\
 |    _________              /  2    2       2    |             /  2    2 |
 |   /  2    2           x*\/  r  - x     I*r *log\-2*x + 2*I*\/  r  - x  /
 | \/  r  - x   dx = C + -------------- - ---------------------------------
 |                             2                          2                
/                                                                          

$$\int \sqrt{r^{2} - x^{2}}\, dx = C - \frac{i r^{2} \log{\left(- 2 x + 2 i \sqrt{r^{2} - x^{2}} \right)}}{2} + \frac{x \sqrt{r^{2} - x^{2}}}{2}$$