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Resolución de integrales/ t^2-1/ sqrt(t^2-1)/t^3

Integral de sqrt(t^2-1)/t^3 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 -3               
  /               
 |                
 |     ________   
 |    /  2        
 |  \/  t  - 1    
 |  ----------- dt
 |        3       
 |       t        
 |                
/                 
-2
$$\int\limits_{-2}^{-3} \frac{\sqrt{t^{2} - 1}}{t^{3}}\, dt$$ 
Integral(sqrt(t^2 - 1)/t^3, (t, -2, -3))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=sec(_theta), rewritten=sin(_theta)**2, substep=RewriteRule(rewritten=1/2 - cos(2*_theta)/2, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_theta), ConstantTimesRule(constant=-1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/2, symbol=_theta)], context=1/2 - cos(2*_theta)/2, symbol=_theta), context=sin(_theta)**2, symbol=_theta), restriction=(t > -1) & (t < 1), context=sqrt(t**2 - 1)/t**3, symbol=t)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                                       
 |                                                                        
 |    ________          //               ________                        \
 |   /  2               ||              /     1                          |
 | \/  t  - 1           ||    /1\      /  1 - --                         |
 | ----------- dt = C + | -1, t < 1)|
 |                      \\   2           2*t                             /
/
$$\int \frac{\sqrt{t^{2} - 1}}{t^{3}}\, dt = C + \begin{cases} \frac{\operatorname{acos}{\left(\frac{1}{t} \right)}}{2} - \frac{\sqrt{1 - \frac{1}{t^{2}}}}{2 t} & \text{for}\: t > -1 \wedge t < 1 \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
  -2                                                                                             
   /                                                                                             
  |                                                                                              
  |  /                                                               pi*I                        
  |  |      I               I*t               I                   I*e                   1        
  |  |-------------- + ------------- - --------------- - -------------------------  for -- > 1   
  |  |      ________             3/2               3/2              ______________       2       
  |  | 3   /      2      /     2\          /     2\                /       2*pi*I       t        
  |  |t *\/  1 - t     2*\1 - t /      2*t*\1 - t /         2     /       e                      
  |  |                                                   2*t *   /   -1 + -------                
  |  |                                                          /             2                  
  |  |                                                        \/             t                   
- |  <                                                                                         dt
  |  |               _________                                                                   
  |  |              /       2                                                                    
  |  |            \/  -1 + t            1                   1                                    
  |  |            ------------ - ---------------- - ------------------              otherwise    
  |  |                  3               _________             ________                           
  |  |                 t               /       2       2     /     1                             
  |  |                           2*t*\/  -1 + t     2*t *   /  1 - --                            
  |  |                                                     /        2                            
  |  \                                                   \/        t                             
  |                                                                                              
 /                                                                                               
 -3
$$- \int\limits_{-3}^{-2} \begin{cases} \frac{i t}{2 \left(1 - t^{2}\right)^{\frac{3}{2}}} - \frac{i}{2 t \left(1 - t^{2}\right)^{\frac{3}{2}}} - \frac{i e^{i \pi}}{2 t^{2} \sqrt{-1 + \frac{e^{2 i \pi}}{t^{2}}}} + \frac{i}{t^{3} \sqrt{1 - t^{2}}} & \text{for}\: \frac{1}{t^{2}} > 1 \\- \frac{1}{2 t \sqrt{t^{2} - 1}} - \frac{1}{2 t^{2} \sqrt{1 - \frac{1}{t^{2}}}} + \frac{\sqrt{t^{2} - 1}}{t^{3}} & \text{otherwise} \end{cases}\, dt$$ 
=
= 
  -2                                                                                             
   /                                                                                             
  |                                                                                              
  |  /                                                               pi*I                        
  |  |      I               I*t               I                   I*e                   1        
  |  |-------------- + ------------- - --------------- - -------------------------  for -- > 1   
  |  |      ________             3/2               3/2              ______________       2       
  |  | 3   /      2      /     2\          /     2\                /       2*pi*I       t        
  |  |t *\/  1 - t     2*\1 - t /      2*t*\1 - t /         2     /       e                      
  |  |                                                   2*t *   /   -1 + -------                
  |  |                                                          /             2                  
  |  |                                                        \/             t                   
- |  <                                                                                         dt
  |  |               _________                                                                   
  |  |              /       2                                                                    
  |  |            \/  -1 + t            1                   1                                    
  |  |            ------------ - ---------------- - ------------------              otherwise    
  |  |                  3               _________             ________                           
  |  |                 t               /       2       2     /     1                             
  |  |                           2*t*\/  -1 + t     2*t *   /  1 - --                            
  |  |                                                     /        2                            
  |  \                                                   \/        t                             
  |                                                                                              
 /                                                                                               
 -3
$$- \int\limits_{-3}^{-2} \begin{cases} \frac{i t}{2 \left(1 - t^{2}\right)^{\frac{3}{2}}} - \frac{i}{2 t \left(1 - t^{2}\right)^{\frac{3}{2}}} - \frac{i e^{i \pi}}{2 t^{2} \sqrt{-1 + \frac{e^{2 i \pi}}{t^{2}}}} + \frac{i}{t^{3} \sqrt{1 - t^{2}}} & \text{for}\: \frac{1}{t^{2}} > 1 \\- \frac{1}{2 t \sqrt{t^{2} - 1}} - \frac{1}{2 t^{2} \sqrt{1 - \frac{1}{t^{2}}}} + \frac{\sqrt{t^{2} - 1}}{t^{3}} & \text{otherwise} \end{cases}\, dt$$ 
-Integral(Piecewise((i/(t^3*sqrt(1 - t^2)) + i*t/(2*(1 - t^2)^(3/2)) - i/(2*t*(1 - t^2)^(3/2)) - i*exp_polar(pi*i)/(2*t^2*sqrt(-1 + exp_polar(2*pi*i)/t^2)), t^(-2) > 1), (sqrt(-1 + t^2)/t^3 - 1/(2*t*sqrt(-1 + t^2)) - 1/(2*t^2*sqrt(1 - 1/t^2)), True)), (t, -3, -2))
Respuesta numérica [src] 
0.151252443754521
0.151252443754521

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

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