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Integral de 1÷(10-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1           
  /           
 |            
 |     1      
 |  ------- dx
 |        2   
 |  10 - x    
 |            
/             
0             
$$\int\limits_{0}^{1} \frac{1}{10 - x^{2}}\, dx$$
Integral(1/(10 - x^2), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-1, c=10, context=1/(10 - x**2), symbol=x), False), (ArccothRule(a=1, b=-1, c=10, context=1/(10 - x**2), symbol=x), x**2 > 10), (ArctanhRule(a=1, b=-1, c=10, context=1/(10 - x**2), symbol=x), x**2 < 10)], context=1/(10 - x**2), symbol=x)

  1. Añadimos la constante de integración:


Respuesta:

Respuesta (Indefinida) [src]
                    //            /    ____\             \
                    ||  ____      |x*\/ 10 |             |
                    ||\/ 10 *acoth|--------|             |
  /                 ||            \   10   /       2     |
 |                  ||----------------------  for x  > 10|
 |    1             ||          10                       |
 | ------- dx = C + |<                                   |
 |       2          ||            /    ____\             |
 | 10 - x           ||  ____      |x*\/ 10 |             |
 |                  ||\/ 10 *atanh|--------|             |
/                   ||            \   10   /       2     |
                    ||----------------------  for x  < 10|
                    \\          10                       /
$$\int \frac{1}{10 - x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{10} \operatorname{acoth}{\left(\frac{\sqrt{10} x}{10} \right)}}{10} & \text{for}\: x^{2} > 10 \\\frac{\sqrt{10} \operatorname{atanh}{\left(\frac{\sqrt{10} x}{10} \right)}}{10} & \text{for}\: x^{2} < 10 \end{cases}$$
Gráfica
Respuesta [src]
    ____ /          /       ____\\     ____    /  ____\     ____ /          /  ____\\     ____    /      ____\
  \/ 10 *\pi*I + log\-1 + \/ 10 //   \/ 10 *log\\/ 10 /   \/ 10 *\pi*I + log\\/ 10 //   \/ 10 *log\1 + \/ 10 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
                 20                          20                        20                         20          
$$- \frac{\sqrt{10} \log{\left(\sqrt{10} \right)}}{20} + \frac{\sqrt{10} \log{\left(1 + \sqrt{10} \right)}}{20} - \frac{\sqrt{10} \left(\log{\left(-1 + \sqrt{10} \right)} + i \pi\right)}{20} + \frac{\sqrt{10} \left(\log{\left(\sqrt{10} \right)} + i \pi\right)}{20}$$
=
=
    ____ /          /       ____\\     ____    /  ____\     ____ /          /  ____\\     ____    /      ____\
  \/ 10 *\pi*I + log\-1 + \/ 10 //   \/ 10 *log\\/ 10 /   \/ 10 *\pi*I + log\\/ 10 //   \/ 10 *log\1 + \/ 10 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
                 20                          20                        20                         20          
$$- \frac{\sqrt{10} \log{\left(\sqrt{10} \right)}}{20} + \frac{\sqrt{10} \log{\left(1 + \sqrt{10} \right)}}{20} - \frac{\sqrt{10} \left(\log{\left(-1 + \sqrt{10} \right)} + i \pi\right)}{20} + \frac{\sqrt{10} \left(\log{\left(\sqrt{10} \right)} + i \pi\right)}{20}$$
-sqrt(10)*(pi*i + log(-1 + sqrt(10)))/20 - sqrt(10)*log(sqrt(10))/20 + sqrt(10)*(pi*i + log(sqrt(10)))/20 + sqrt(10)*log(1 + sqrt(10))/20
Respuesta numérica [src]
0.103548829491406
0.103548829491406

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