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Resolución de integrales/ sqrt(((a1+b1)^2)*t+((a2+b2)^2)*t+((a3+b3)^2)*t)

Integral de sqrt(((a1+b1)^2)*t+((a2+b2)^2)*t+((a3+b3)^2)*t) dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  t                                                   
  /                                                   
 |                                                    
 |     ____________________________________________   
 |    /          2              2              2      
 |  \/  (a1 + b1) *t + (a2 + b2) *t + (a3 + b3) *t  dt
 |                                                    
/                                                     
0
∫0tt(a3+b3)2+(t(a1+b1)2+t(a2+b2)2) dt\int\limits_{0}^{t} \sqrt{t \left(a_{3} + b_{3}\right)^{2} + \left(t \left(a_{1} + b_{1}\right)^{2} + t \left(a_{2} + b_{2}\right)^{2}\right)}\, dt 
Integral(sqrt((a1 + b1)^2*t + (a2 + b2)^2*t + (a3 + b3)^2*t), (t, 0, t))
Respuesta (Indefinida) [src] 
                                                            //                                                       3/2                                                                                   \
                                                            ||           /         2              2              2  \                                                                                      |
  /                                                         ||         2*\(a1 + b1) *t + (a2 + b2) *t + (a3 + b3) *t/                    2     2     2     2     2     2                                   |
 |                                                          ||-------------------------------------------------------------------  for a1  + a2  + a3  + b1  + b2  + b3  + 2*a1*b1 + 2*a2*b2 + 2*a3*b3 != 0|
 |    ____________________________________________          ||  /  2     2     2     2     2     2                              \                                                                          |
 |   /          2              2              2             ||3*\a1  + a2  + a3  + b1  + b2  + b3  + 2*a1*b1 + 2*a2*b2 + 2*a3*b3/                                                                          |
 | \/  (a1 + b1) *t + (a2 + b2) *t + (a3 + b3) *t  dt = C + |<                                                                                                                                             |
 |                                                          ||                                                       3/2                                                                                   |
/                                                           ||           /         2              2              2  \                                                                                      |
                                                            ||         2*\(a1 + b1) *t + (a2 + b2) *t + (a3 + b3) *t/                                                                                      |
                                                            ||         -------------------------------------------------                                          otherwise                                |
                                                            ||                /         2            2            2\                                                                                       |
                                                            \\              3*\(a1 + b1)  + (a2 + b2)  + (a3 + b3) /                                                                                       /
∫t(a3+b3)2+(t(a1+b1)2+t(a2+b2)2) dt=C+{2(t(a3+b3)2+(t(a1+b1)2+t(a2+b2)2))323(a12+2a1b1+a22+2a2b2+a32+2a3b3+b12+b22+b32)for a12+2a1b1+a22+2a2b2+a32+2a3b3+b12+b22+b32≠02(t(a3+b3)2+(t(a1+b1)2+t(a2+b2)2))323((a1+b1)2+(a2+b2)2+(a3+b3)2)otherwise\int \sqrt{t \left(a_{3} + b_{3}\right)^{2} + \left(t \left(a_{1} + b_{1}\right)^{2} + t \left(a_{2} + b_{2}\right)^{2}\right)}\, dt = C + \begin{cases} \frac{2 \left(t \left(a_{3} + b_{3}\right)^{2} + \left(t \left(a_{1} + b_{1}\right)^{2} + t \left(a_{2} + b_{2}\right)^{2}\right)\right)^{\frac{3}{2}}}{3 \left(a_{1}^{2} + 2 a_{1} b_{1} + a_{2}^{2} + 2 a_{2} b_{2} + a_{3}^{2} + 2 a_{3} b_{3} + b_{1}^{2} + b_{2}^{2} + b_{3}^{2}\right)} & \text{for}\: a_{1}^{2} + 2 a_{1} b_{1} + a_{2}^{2} + 2 a_{2} b_{2} + a_{3}^{2} + 2 a_{3} b_{3} + b_{1}^{2} + b_{2}^{2} + b_{3}^{2} \neq 0 \\\frac{2 \left(t \left(a_{3} + b_{3}\right)^{2} + \left(t \left(a_{1} + b_{1}\right)^{2} + t \left(a_{2} + b_{2}\right)^{2}\right)\right)^{\frac{3}{2}}}{3 \left(\left(a_{1} + b_{1}\right)^{2} + \left(a_{2} + b_{2}\right)^{2} + \left(a_{3} + b_{3}\right)^{2}\right)} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
                                              3/2
  /           2              2              2\   
2*\t*(a1 + b1)  + t*(a2 + b2)  + t*(a3 + b3) /   
-------------------------------------------------
       /         2            2            2\    
     3*\(a1 + b1)  + (a2 + b2)  + (a3 + b3) /
2(t(a1+b1)2+t(a2+b2)2+t(a3+b3)2)323((a1+b1)2+(a2+b2)2+(a3+b3)2)\frac{2 \left(t \left(a_{1} + b_{1}\right)^{2} + t \left(a_{2} + b_{2}\right)^{2} + t \left(a_{3} + b_{3}\right)^{2}\right)^{\frac{3}{2}}}{3 \left(\left(a_{1} + b_{1}\right)^{2} + \left(a_{2} + b_{2}\right)^{2} + \left(a_{3} + b_{3}\right)^{2}\right)} 
=
= 
                                              3/2
  /           2              2              2\   
2*\t*(a1 + b1)  + t*(a2 + b2)  + t*(a3 + b3) /   
-------------------------------------------------
       /         2            2            2\    
     3*\(a1 + b1)  + (a2 + b2)  + (a3 + b3) /
2(t(a1+b1)2+t(a2+b2)2+t(a3+b3)2)323((a1+b1)2+(a2+b2)2+(a3+b3)2)\frac{2 \left(t \left(a_{1} + b_{1}\right)^{2} + t \left(a_{2} + b_{2}\right)^{2} + t \left(a_{3} + b_{3}\right)^{2}\right)^{\frac{3}{2}}}{3 \left(\left(a_{1} + b_{1}\right)^{2} + \left(a_{2} + b_{2}\right)^{2} + \left(a_{3} + b_{3}\right)^{2}\right)} 
2*(t*(a1 + b1)^2 + t*(a2 + b2)^2 + t*(a3 + b3)^2)^(3/2)/(3*((a1 + b1)^2 + (a2 + b2)^2 + (a3 + b3)^2))

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

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