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Resolución de integrales/ sqrt(1+(y+1)^2)

Integral de sqrt(1+(y+1)^2) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
      ___                    
  5*\/ 5                     
     /                       
    |                        
    |       ______________   
    |      /            2    
    |    \/  1 + (y + 1)   dy
    |                        
   /                         
     ___                     
-5*\/ 5
∫−5555(y+1)2+1 dy\int\limits_{- 5 \sqrt{5}}^{5 \sqrt{5}} \sqrt{\left(y + 1\right)^{2} + 1}\, dy 
Integral(sqrt(1 + (y + 1)^2), (y, -5*sqrt(5), 5*sqrt(5)))
Respuesta (Indefinida) [src] 
  /                                                                   
 |                                              ______________        
 |    ______________                           /            2         
 |   /            2           asinh(1 + y)   \/  1 + (1 + y)  *(1 + y)
 | \/  1 + (y + 1)   dy = C + ------------ + -------------------------
 |                                 2                     2            
/
∫(y+1)2+1 dy=C+(y+1)(y+1)2+12+asinh⁡(y+1)2\int \sqrt{\left(y + 1\right)^{2} + 1}\, dy = C + \frac{\left(y + 1\right) \sqrt{\left(y + 1\right)^{2} + 1}}{2} + \frac{\operatorname{asinh}{\left(y + 1 \right)}}{2}
Simplificar
Gráfica
02468-10-8-6-4-210-100100
Trazar el gráfico f(x)
Respuesta [src] 
                         ____________________                     ____________________                                                  
                        /                  2                     /                  2                                                   
     /        ___\     /      /        ___\   /        ___\     /      /        ___\   /        ___\   /        ___\      /         ___\
asinh\1 + 5*\/ 5 /   \/   1 + \1 + 5*\/ 5 /  *\1 + 5*\/ 5 /   \/   1 + \1 - 5*\/ 5 /  *\1 - 5*\/ 5 /   \1 - 5*\/ 5 /*asinh\-1 + 5*\/ 5 /
------------------ + -------------------------------------- - -------------------------------------- - ---------------------------------
        2                              2                                        2                                 /         ___\        
                                                                                                                2*\-1 + 5*\/ 5 /
−(1−55)asinh⁡(−1+55)2(−1+55)+asinh⁡(1+55)2−(1−55)1+(1−55)22+(1+55)1+(1+55)22- \frac{\left(1 - 5 \sqrt{5}\right) \operatorname{asinh}{\left(-1 + 5 \sqrt{5} \right)}}{2 \left(-1 + 5 \sqrt{5}\right)} + \frac{\operatorname{asinh}{\left(1 + 5 \sqrt{5} \right)}}{2} - \frac{\left(1 - 5 \sqrt{5}\right) \sqrt{1 + \left(1 - 5 \sqrt{5}\right)^{2}}}{2} + \frac{\left(1 + 5 \sqrt{5}\right) \sqrt{1 + \left(1 + 5 \sqrt{5}\right)^{2}}}{2} 
=
= 
                         ____________________                     ____________________                                                  
                        /                  2                     /                  2                                                   
     /        ___\     /      /        ___\   /        ___\     /      /        ___\   /        ___\   /        ___\      /         ___\
asinh\1 + 5*\/ 5 /   \/   1 + \1 + 5*\/ 5 /  *\1 + 5*\/ 5 /   \/   1 + \1 - 5*\/ 5 /  *\1 - 5*\/ 5 /   \1 - 5*\/ 5 /*asinh\-1 + 5*\/ 5 /
------------------ + -------------------------------------- - -------------------------------------- - ---------------------------------
        2                              2                                        2                                 /         ___\        
                                                                                                                2*\-1 + 5*\/ 5 /
−(1−55)asinh⁡(−1+55)2(−1+55)+asinh⁡(1+55)2−(1−55)1+(1−55)22+(1+55)1+(1+55)22- \frac{\left(1 - 5 \sqrt{5}\right) \operatorname{asinh}{\left(-1 + 5 \sqrt{5} \right)}}{2 \left(-1 + 5 \sqrt{5}\right)} + \frac{\operatorname{asinh}{\left(1 + 5 \sqrt{5} \right)}}{2} - \frac{\left(1 - 5 \sqrt{5}\right) \sqrt{1 + \left(1 - 5 \sqrt{5}\right)^{2}}}{2} + \frac{\left(1 + 5 \sqrt{5}\right) \sqrt{1 + \left(1 + 5 \sqrt{5}\right)^{2}}}{2} 
asinh(1 + 5*sqrt(5))/2 + sqrt(1 + (1 + 5*sqrt(5))^2)*(1 + 5*sqrt(5))/2 - sqrt(1 + (1 - 5*sqrt(5))^2)*(1 - 5*sqrt(5))/2 - (1 - 5*sqrt(5))*asinh(-1 + 5*sqrt(5))/(2*(-1 + 5*sqrt(5)))
Respuesta numérica [src] 
129.604310130225
129.604310130225

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

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