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Resolución de integrales/ a-x^2/ sqrt(a-x^2)

Integral de sqrt(a-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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∫01a−x2 dx\int\limits_{0}^{1} \sqrt{a - x^{2}}\, dx 
Integral(sqrt(a - x^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                        //           /  x  \                                                          \
                        ||  I*a*acosh|-----|                                                          |
                        ||           |  ___|               3                     ___          | 2|    |
                        ||           \\/ a /            I*x                I*x*\/ a           |x |    |
                        ||- ---------------- + ---------------------- - ----------------  for |--| > 1|
  /                     ||         2                        _________          _________      |a |    |
 |                      ||                                 /       2          /       2               |
 |    ________          ||                         ___    /       x          /       x                |
 |   /      2           ||                     2*\/ a *  /   -1 + --    2*  /   -1 + --               |
 | \/  a - x   dx = C + |<                             \/         a       \/         a                |
 |                      ||                                                                            |
/                       ||                                         ________                           |
                        ||                  /  x  \               /      2                            |
                        ||            a*asin|-----|       ___    /      x                             |
                        ||                  |  ___|   x*\/ a *  /   1 - --                            |
                        ||                  \\/ a /           \/        a                             |
                        ||            ------------- + ---------------------                otherwise  |
                        ||                  2                   2                                     |
                        \\                                                                            /
∫a−x2 dx=C+{−iax2−1+x2a−iaacosh⁡(xa)2+ix32a−1+x2afor ∣x2a∣>1ax1−x2a2+aasin⁡(xa)2otherwise\int \sqrt{a - x^{2}}\, dx = C + \begin{cases} - \frac{i \sqrt{a} x}{2 \sqrt{-1 + \frac{x^{2}}{a}}} - \frac{i a \operatorname{acosh}{\left(\frac{x}{\sqrt{a}} \right)}}{2} + \frac{i x^{3}}{2 \sqrt{a} \sqrt{-1 + \frac{x^{2}}{a}}} & \text{for}\: \left|{\frac{x^{2}}{a}}\right| > 1 \\\frac{\sqrt{a} x \sqrt{1 - \frac{x^{2}}{a}}}{2} + \frac{a \operatorname{asin}{\left(\frac{x}{\sqrt{a}} \right)}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
  1                                                                                                        
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 |  /         ___                  2                      4                       2                2       
 |  |     I*\/ a                I*x                    I*x                   3*I*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                                                                         
 |  |            \/ a *  /   1 - --           ___                   2                                      
 |  |                  \/        a          \/ a                   x                                       
 |  |            ------------------- + --------------- - ---------------------                otherwise    
 |  |                     2                   ________                ________                             
 |  |                                        /      2                /      2                              
 |  |                                       /      x         ___    /      x                               
 |  |                                  2*  /   1 - --    2*\/ a *  /   1 - --                              
 |  |                                    \/        a             \/        a                               
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∫01{−ia−1+x2a+3ix22a−1+x2a+ix22a(−1+x2a)32−ix42a32(−1+x2a)32for x2∣a∣>1a1−x2a2+a21−x2a−x22a1−x2aotherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{i \sqrt{a}}{\sqrt{-1 + \frac{x^{2}}{a}}} + \frac{3 i x^{2}}{2 \sqrt{a} \sqrt{-1 + \frac{x^{2}}{a}}} + \frac{i x^{2}}{2 \sqrt{a} \left(-1 + \frac{x^{2}}{a}\right)^{\frac{3}{2}}} - \frac{i 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{\sqrt{a} \sqrt{1 - \frac{x^{2}}{a}}}{2} + \frac{\sqrt{a}}{2 \sqrt{1 - \frac{x^{2}}{a}}} - \frac{x^{2}}{2 \sqrt{a} \sqrt{1 - \frac{x^{2}}{a}}} & \text{otherwise} \end{cases}\, dx 
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  1                                                                                                        
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 |  /         ___                  2                      4                       2                2       
 |  |     I*\/ a                I*x                    I*x                   3*I*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                                                                         
 |  |            \/ a *  /   1 - --           ___                   2                                      
 |  |                  \/        a          \/ a                   x                                       
 |  |            ------------------- + --------------- - ---------------------                otherwise    
 |  |                     2                   ________                ________                             
 |  |                                        /      2                /      2                              
 |  |                                       /      x         ___    /      x                               
 |  |                                  2*  /   1 - --    2*\/ a *  /   1 - --                              
 |  |                                    \/        a             \/        a                               
 |  \                                                                                                      
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/                                                                                                          
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∫01{−ia−1+x2a+3ix22a−1+x2a+ix22a(−1+x2a)32−ix42a32(−1+x2a)32for x2∣a∣>1a1−x2a2+a21−x2a−x22a1−x2aotherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{i \sqrt{a}}{\sqrt{-1 + \frac{x^{2}}{a}}} + \frac{3 i x^{2}}{2 \sqrt{a} \sqrt{-1 + \frac{x^{2}}{a}}} + \frac{i x^{2}}{2 \sqrt{a} \left(-1 + \frac{x^{2}}{a}\right)^{\frac{3}{2}}} - \frac{i 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{\sqrt{a} \sqrt{1 - \frac{x^{2}}{a}}}{2} + \frac{\sqrt{a}}{2 \sqrt{1 - \frac{x^{2}}{a}}} - \frac{x^{2}}{2 \sqrt{a} \sqrt{1 - \frac{x^{2}}{a}}} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((-i*sqrt(a)/sqrt(-1 + x^2/a) + i*x^2/(2*sqrt(a)*(-1 + x^2/a)^(3/2)) - i*x^4/(2*a^(3/2)*(-1 + x^2/a)^(3/2)) + 3*i*x^2/(2*sqrt(a)*sqrt(-1 + x^2/a)), x^2/|a| > 1), (sqrt(a)*sqrt(1 - x^2/a)/2 + sqrt(a)/(2*sqrt(1 - x^2/a)) - 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.

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