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Integral de 1/(√4-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1              
  /              
 |               
 |      1        
 |  ---------- dx
 |    ___    2   
 |  \/ 4  - x    
 |               
/                
0                
$$\int\limits_{0}^{1} \frac{1}{- x^{2} + \sqrt{4}}\, dx$$
Integral(1/(sqrt(4) - x^2), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-1, c=sqrt(4), context=1/(-x**2 + sqrt(4)), symbol=x), False), (ArccothRule(a=1, b=-1, c=sqrt(4), context=1/(-x**2 + sqrt(4)), symbol=x), x**2 > 2), (ArctanhRule(a=1, b=-1, c=sqrt(4), context=1/(-x**2 + sqrt(4)), symbol=x), x**2 < 2)], context=1/(-x**2 + sqrt(4)), symbol=x)

  1. Añadimos la constante de integración:


Respuesta:

Respuesta (Indefinida) [src]
                       //           /    ___\            \
                       ||  ___      |x*\/ 2 |            |
                       ||\/ 2 *acoth|-------|            |
  /                    ||           \   2   /       2    |
 |                     ||--------------------  for x  > 2|
 |     1               ||         2                      |
 | ---------- dx = C + |<                                |
 |   ___    2          ||           /    ___\            |
 | \/ 4  - x           ||  ___      |x*\/ 2 |            |
 |                     ||\/ 2 *atanh|-------|            |
/                      ||           \   2   /       2    |
                       ||--------------------  for x  < 2|
                       \\         2                      /
$$\int \frac{1}{- x^{2} + \sqrt{4}}\, dx = C + \begin{cases} \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} > 2 \\\frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} < 2 \end{cases}$$
Gráfica
Respuesta [src]
    ___ /          /       ___\\     ___    /  ___\     ___ /          /  ___\\     ___    /      ___\
  \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /   \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
                4                         4                       4                        4          
$$- \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{4} - \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{4}$$
=
=
    ___ /          /       ___\\     ___    /  ___\     ___ /          /  ___\\     ___    /      ___\
  \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /   \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
                4                         4                       4                        4          
$$- \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{4} - \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{4}$$
-sqrt(2)*(pi*i + log(-1 + sqrt(2)))/4 - sqrt(2)*log(sqrt(2))/4 + sqrt(2)*(pi*i + log(sqrt(2)))/4 + sqrt(2)*log(1 + sqrt(2))/4
Respuesta numérica [src]
0.623225240140231
0.623225240140231

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