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Resolución de integrales/ (2*e^(1-((2arsinx)/pi))/pi*sqrt(1-x^2))

Integral de (2*e^(1-((2arsinx)/pi))/pi*sqrt(1-x^2)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                                
  /                                
 |                                 
 |         2*asin(x)               
 |     1 - ---------               
 |             pi       ________   
 |  2*E                /      2    
 |  ----------------*\/  1 - x   dx
 |         pi                      
 |                                 
/                                  
0
∫012e−2asin⁡(x)π+1π1−x2 dx\int\limits_{0}^{1} \frac{2 e^{- \frac{2 \operatorname{asin}{\left(x \right)}}{\pi} + 1}}{\pi} \sqrt{1 - x^{2}}\, dx 
Integral(((2*E^(1 - 2*asin(x)/pi))/pi)*sqrt(1 - x^2), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    2e−2asin⁡(x)π+1π1−x2=2e1−x2e−2asin⁡(x)ππ\frac{2 e^{- \frac{2 \operatorname{asin}{\left(x \right)}}{\pi} + 1}}{\pi} \sqrt{1 - x^{2}} = \frac{2 e \sqrt{1 - x^{2}} e^{- \frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}}{\pi}

  2. La integral del producto de una función por una constante es la constante por la integral de esta función:

    ∫2e1−x2e−2asin⁡(x)ππ dx=2e∫1−x2e−2asin⁡(x)π dxπ\int \frac{2 e \sqrt{1 - x^{2}} e^{- \frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}}{\pi}\, dx = \frac{2 e \int \sqrt{1 - x^{2}} e^{- \frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}\, dx}{\pi}

    1. No puedo encontrar los pasos en la búsqueda de esta integral.

      Pero la integral

      2πx24e2asin⁡(x)π+4π2e2asin⁡(x)π+2π2x1−x24e2asin⁡(x)π+4π2e2asin⁡(x)π−π34e2asin⁡(x)π+4π2e2asin⁡(x)π−2π4e2asin⁡(x)π+4π2e2asin⁡(x)π\frac{2 \pi x^{2}}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}} + \frac{2 \pi^{2} x \sqrt{1 - x^{2}}}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}} - \frac{\pi^{3}}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}} - \frac{2 \pi}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}}

    Por lo tanto, el resultado es: 2e(2πx24e2asin⁡(x)π+4π2e2asin⁡(x)π+2π2x1−x24e2asin⁡(x)π+4π2e2asin⁡(x)π−π34e2asin⁡(x)π+4π2e2asin⁡(x)π−2π4e2asin⁡(x)π+4π2e2asin⁡(x)π)π\frac{2 e \left(\frac{2 \pi x^{2}}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}} + \frac{2 \pi^{2} x \sqrt{1 - x^{2}}}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}} - \frac{\pi^{3}}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}} - \frac{2 \pi}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}}\right)}{\pi}

  3. Ahora simplificar:

    (x2+πx1−x2−π22−1)e−2asin⁡(x)π+11+π2\frac{\left(x^{2} + \pi x \sqrt{1 - x^{2}} - \frac{\pi^{2}}{2} - 1\right) e^{- \frac{2 \operatorname{asin}{\left(x \right)}}{\pi} + 1}}{1 + \pi^{2}}

  4. Añadimos la constante de integración:

    (x2+πx1−x2−π22−1)e−2asin⁡(x)π+11+π2+constant\frac{\left(x^{2} + \pi x \sqrt{1 - x^{2}} - \frac{\pi^{2}}{2} - 1\right) e^{- \frac{2 \operatorname{asin}{\left(x \right)}}{\pi} + 1}}{1 + \pi^{2}}+ \mathrm{constant}

Respuesta:

(x2+πx1−x2−π22−1)e−2asin⁡(x)π+11+π2+constant\frac{\left(x^{2} + \pi x \sqrt{1 - x^{2}} - \frac{\pi^{2}}{2} - 1\right) e^{- \frac{2 \operatorname{asin}{\left(x \right)}}{\pi} + 1}}{1 + \pi^{2}}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                             /                                                                                                                         ________      \
                                             |                  3                                                                     2                           2   /      2       |
  /                                          |                pi                                2*pi                            2*pi*x                      2*x*pi *\/  1 - x        |
 |                                       2*E*|- ------------------------------- - ------------------------------- + ------------------------------- + -------------------------------|
 |        2*asin(x)                          |     2*asin(x)          2*asin(x)      2*asin(x)          2*asin(x)      2*asin(x)          2*asin(x)      2*asin(x)          2*asin(x)|
 |    1 - ---------                          |     ---------          ---------      ---------          ---------      ---------          ---------      ---------          ---------|
 |            pi       ________              |         pi          2      pi             pi          2      pi             pi          2      pi             pi          2      pi   |
 | 2*E                /      2               \  4*e          + 4*pi *e            4*e          + 4*pi *e            4*e          + 4*pi *e            4*e          + 4*pi *e         /
 | ----------------*\/  1 - x   dx = C + ---------------------------------------------------------------------------------------------------------------------------------------------
 |        pi                                                                                                   pi                                                                     
 |                                                                                                                                                                                    
/
∫2e−2asin⁡(x)π+1π1−x2 dx=C+2e(2πx24e2asin⁡(x)π+4π2e2asin⁡(x)π+2π2x1−x24e2asin⁡(x)π+4π2e2asin⁡(x)π−π34e2asin⁡(x)π+4π2e2asin⁡(x)π−2π4e2asin⁡(x)π+4π2e2asin⁡(x)π)π\int \frac{2 e^{- \frac{2 \operatorname{asin}{\left(x \right)}}{\pi} + 1}}{\pi} \sqrt{1 - x^{2}}\, dx = C + \frac{2 e \left(\frac{2 \pi x^{2}}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}} + \frac{2 \pi^{2} x \sqrt{1 - x^{2}}}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}} - \frac{\pi^{3}}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}} - \frac{2 \pi}{4 e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}} + 4 \pi^{2} e^{\frac{2 \operatorname{asin}{\left(x \right)}}{\pi}}}\right)}{\pi}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.905-5
Trazar el gráfico f(x)
Respuesta [src] 
    /        3              \                
    |    E*pi        2*E*pi |                
  2*|- --------- - ---------|                
    |          2           2|            2   
    \  4 + 4*pi    4 + 4*pi /      2*E*pi    
- --------------------------- - -------------
               pi                           2
                                4*E + 4*E*pi
−2eπ24e+4eπ2−2(−eπ34+4π2−2eπ4+4π2)π- \frac{2 e \pi^{2}}{4 e + 4 e \pi^{2}} - \frac{2 \left(- \frac{e \pi^{3}}{4 + 4 \pi^{2}} - \frac{2 e \pi}{4 + 4 \pi^{2}}\right)}{\pi} 
=
= 
    /        3              \                
    |    E*pi        2*E*pi |                
  2*|- --------- - ---------|                
    |          2           2|            2   
    \  4 + 4*pi    4 + 4*pi /      2*E*pi    
- --------------------------- - -------------
               pi                           2
                                4*E + 4*E*pi
−2eπ24e+4eπ2−2(−eπ34+4π2−2eπ4+4π2)π- \frac{2 e \pi^{2}}{4 e + 4 e \pi^{2}} - \frac{2 \left(- \frac{e \pi^{3}}{4 + 4 \pi^{2}} - \frac{2 e \pi}{4 + 4 \pi^{2}}\right)}{\pi} 
-2*(-E*pi^3/(4 + 4*pi^2) - 2*E*pi/(4 + 4*pi^2))/pi - 2*E*pi^2/(4*E + 4*E*pi^2)
Respuesta numérica [src] 
1.03018126175525
1.03018126175525

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

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