Sr Examen

Otras calculadoras

Resolución de integrales/ 1/2dx/ (x^2-a)^1/2dx

Integral de (x^2-a)^1/2dx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |     ________   
 |    /  2        
 |  \/  x  - a  dx
 |                
/                 
0
∫01−a+x2 dx\int\limits_{0}^{1} \sqrt{- a + x^{2}}\, dx 
Integral(sqrt(x^2 - a), (x, 0, 1))
Respuesta (Indefinida) [src] 
                        //         /  x  \                                                          \
                        ||  a*acosh|-----|                                                          |
                        ||         |  ___|              3                     ___           | 2|    |
                        ||         \\/ a /             x                  x*\/ a            |x |    |
                        ||- -------------- + ---------------------- - ----------------  for |--| > 1|
  /                     ||        2                       _________          _________      |a |    |
 |                      ||                               /       2          /       2               |
 |    ________          ||                       ___    /       x          /       x                |
 |   /  2               ||                   2*\/ a *  /   -1 + --    2*  /   -1 + --               |
 | \/  x  - a  dx = C + |<                           \/         a       \/         a                |
 |                      ||                                                                          |
/                       ||                                          ________                        |
                        ||                 /  x  \                 /      2                         |
                        ||         I*a*asin|-----|         ___    /      x                          |
                        ||                 |  ___|   I*x*\/ a *  /   1 - --                         |
                        ||                 \\/ a /             \/        a                          |
                        ||         --------------- + -----------------------             otherwise  |
                        ||                2                     2                                   |
                        \\                                                                          /
∫−a+x2 dx=C+{−ax2−1+x2a−aacosh⁡(xa)2+x32a−1+x2afor ∣x2a∣>1iax1−x2a2+iaasin⁡(xa)2otherwise\int \sqrt{- a + x^{2}}\, dx = C + \begin{cases} - \frac{\sqrt{a} x}{2 \sqrt{-1 + \frac{x^{2}}{a}}} - \frac{a \operatorname{acosh}{\left(\frac{x}{\sqrt{a}} \right)}}{2} + \frac{x^{3}}{2 \sqrt{a} \sqrt{-1 + \frac{x^{2}}{a}}} & \text{for}\: \left|{\frac{x^{2}}{a}}\right| > 1 \\\frac{i \sqrt{a} x \sqrt{1 - \frac{x^{2}}{a}}}{2} + \frac{i a \operatorname{asin}{\left(\frac{x}{\sqrt{a}} \right)}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
  1                                                                                                        
  /                                                                                                        
 |                                                                                                         
 |  /        ___                  2                      4                       2                 2       
 |  |      \/ a                  x                      x                     3*x                 x        
 |  |- -------------- + -------------------- - ------------------- + ----------------------  for --- > 1   
 |  |       _________                    3/2                   3/2                _________      |a|       
 |  |      /       2            /      2\             /      2\                  /       2                 
 |  |     /       x         ___ |     x |         3/2 |     x |          ___    /       x                  
 |  |    /   -1 + --    2*\/ a *|-1 + --|      2*a   *|-1 + --|      2*\/ a *  /   -1 + --                 
 |  |  \/         a             \     a /             \     a /              \/         a                  
 |  |                                                                                                      
 |  |                        ________                                                                      
 |  <                       /      2                                                                     dx
 |  |               ___    /      x                                                                        
 |  |           I*\/ a *  /   1 - --            ___                   2                                    
 |  |                   \/        a         I*\/ a                 I*x                                     
 |  |           --------------------- + --------------- - ---------------------               otherwise    
 |  |                     2                    ________                ________                            
 |  |                                         /      2                /      2                             
 |  |                                        /      x         ___    /      x                              
 |  |                                   2*  /   1 - --    2*\/ a *  /   1 - --                             
 |  |                                     \/        a             \/        a                              
 |  \                                                                                                      
 |                                                                                                         
/                                                                                                          
0
∫01{−a−1+x2a+3x22a−1+x2a+x22a(−1+x2a)32−x42a32(−1+x2a)32for x2∣a∣>1ia1−x2a2+ia21−x2a−ix22a1−x2aotherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{\sqrt{a}}{\sqrt{-1 + \frac{x^{2}}{a}}} + \frac{3 x^{2}}{2 \sqrt{a} \sqrt{-1 + \frac{x^{2}}{a}}} + \frac{x^{2}}{2 \sqrt{a} \left(-1 + \frac{x^{2}}{a}\right)^{\frac{3}{2}}} - \frac{x^{4}}{2 a^{\frac{3}{2}} \left(-1 + \frac{x^{2}}{a}\right)^{\frac{3}{2}}} & \text{for}\: \frac{x^{2}}{\left|{a}\right|} > 1 \\\frac{i \sqrt{a} \sqrt{1 - \frac{x^{2}}{a}}}{2} + \frac{i \sqrt{a}}{2 \sqrt{1 - \frac{x^{2}}{a}}} - \frac{i x^{2}}{2 \sqrt{a} \sqrt{1 - \frac{x^{2}}{a}}} & \text{otherwise} \end{cases}\, dx 
=
= 
  1                                                                                                        
  /                                                                                                        
 |                                                                                                         
 |  /        ___                  2                      4                       2                 2       
 |  |      \/ a                  x                      x                     3*x                 x        
 |  |- -------------- + -------------------- - ------------------- + ----------------------  for --- > 1   
 |  |       _________                    3/2                   3/2                _________      |a|       
 |  |      /       2            /      2\             /      2\                  /       2                 
 |  |     /       x         ___ |     x |         3/2 |     x |          ___    /       x                  
 |  |    /   -1 + --    2*\/ a *|-1 + --|      2*a   *|-1 + --|      2*\/ a *  /   -1 + --                 
 |  |  \/         a             \     a /             \     a /              \/         a                  
 |  |                                                                                                      
 |  |                        ________                                                                      
 |  <                       /      2                                                                     dx
 |  |               ___    /      x                                                                        
 |  |           I*\/ a *  /   1 - --            ___                   2                                    
 |  |                   \/        a         I*\/ a                 I*x                                     
 |  |           --------------------- + --------------- - ---------------------               otherwise    
 |  |                     2                    ________                ________                            
 |  |                                         /      2                /      2                             
 |  |                                        /      x         ___    /      x                              
 |  |                                   2*  /   1 - --    2*\/ a *  /   1 - --                             
 |  |                                     \/        a             \/        a                              
 |  \                                                                                                      
 |                                                                                                         
/                                                                                                          
0
∫01{−a−1+x2a+3x22a−1+x2a+x22a(−1+x2a)32−x42a32(−1+x2a)32for x2∣a∣>1ia1−x2a2+ia21−x2a−ix22a1−x2aotherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{\sqrt{a}}{\sqrt{-1 + \frac{x^{2}}{a}}} + \frac{3 x^{2}}{2 \sqrt{a} \sqrt{-1 + \frac{x^{2}}{a}}} + \frac{x^{2}}{2 \sqrt{a} \left(-1 + \frac{x^{2}}{a}\right)^{\frac{3}{2}}} - \frac{x^{4}}{2 a^{\frac{3}{2}} \left(-1 + \frac{x^{2}}{a}\right)^{\frac{3}{2}}} & \text{for}\: \frac{x^{2}}{\left|{a}\right|} > 1 \\\frac{i \sqrt{a} \sqrt{1 - \frac{x^{2}}{a}}}{2} + \frac{i \sqrt{a}}{2 \sqrt{1 - \frac{x^{2}}{a}}} - \frac{i x^{2}}{2 \sqrt{a} \sqrt{1 - \frac{x^{2}}{a}}} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((-sqrt(a)/sqrt(-1 + x^2/a) + x^2/(2*sqrt(a)*(-1 + x^2/a)^(3/2)) - x^4/(2*a^(3/2)*(-1 + x^2/a)^(3/2)) + 3*x^2/(2*sqrt(a)*sqrt(-1 + x^2/a)), x^2/|a| > 1), (i*sqrt(a)*sqrt(1 - x^2/a)/2 + i*sqrt(a)/(2*sqrt(1 - x^2/a)) - i*x^2/(2*sqrt(a)*sqrt(1 - x^2/a)), True)), (x, 0, 1))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

    © Sr Examen – Калькулятор