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Resolución de integrales/ 0-x^2/ 1÷(10-x^2)

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
∫01110−x2 dx\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:

    {10acoth⁡(10x10)10for x2>1010atanh⁡(10x10)10for x2<10+constant\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}+ \mathrm{constant}

Respuesta:

{10acoth⁡(10x10)10for x2>1010atanh⁡(10x10)10for x2<10+constant\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}+ \mathrm{constant}

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                       /
∫110−x2 dx=C+{10acoth⁡(10x10)10for x2>1010atanh⁡(10x10)10for x2<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}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.090.12
Trazar el gráfico f(x)
Respuesta [src] 
    ____ /          /       ____\\     ____    /  ____\     ____ /          /  ____\\     ____    /      ____\
  \/ 10 *\pi*I + log\-1 + \/ 10 //   \/ 10 *log\\/ 10 /   \/ 10 *\pi*I + log\\/ 10 //   \/ 10 *log\1 + \/ 10 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
                 20                          20                        20                         20
−10log⁡(10)20+10log⁡(1+10)20−10(log⁡(−1+10)+iπ)20+10(log⁡(10)+iπ)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
−10log⁡(10)20+10log⁡(1+10)20−10(log⁡(−1+10)+iπ)20+10(log⁡(10)+iπ)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.

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