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Resolución de integrales/ (t-1)/ (-sqrt(t-1)+1)*e^(-p*t)

Integral de (-sqrt(t-1)+1)*e^(-p*t) dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  3                           
  /                           
 |                            
 |  /    _______    \  -p*t   
 |  \- \/ t - 1  + 1/*E     dt
 |                            
/                             
2
∫23e−pt(1−t−1) dt\int\limits_{2}^{3} e^{- p t} \left(1 - \sqrt{t - 1}\right)\, dt 
Integral((-sqrt(t - 1) + 1)*E^((-p)*t), (t, 2, 3))
Respuesta (Indefinida) [src] 
                                               /                                ____     /  ____________\\                            
  /                                   ________ |  ____________  -p*(-1 + t)   \/ pi *erfc\\/ p*(-1 + t) /|  -p   //   t     for p = 0\
 |                                  \/ -1 + t *|\/ p*(-1 + t) *e            + ---------------------------|*e     ||                  |
 | /    _______    \  -p*t                     \                                           2             /       ||  -p*t            |
 | \- \/ t - 1  + 1/*E     dt = C + -------------------------------------------------------------------------- + |<-e                |
 |                                                                   ____________                                ||-------  otherwise|
/                                                                p*\/ p*(-1 + t)                                 ||   p              |
                                                                                                                 \\                  /
∫e−pt(1−t−1) dt=C+{tfor p=0−e−ptpotherwise+t−1(p(t−1)e−p(t−1)+πerfc⁡(p(t−1))2)e−ppp(t−1)\int e^{- p t} \left(1 - \sqrt{t - 1}\right)\, dt = C + \begin{cases} t & \text{for}\: p = 0 \\- \frac{e^{- p t}}{p} & \text{otherwise} \end{cases} + \frac{\sqrt{t - 1} \left(\sqrt{p \left(t - 1\right)} e^{- p \left(t - 1\right)} + \frac{\sqrt{\pi} \operatorname{erfc}{\left(\sqrt{p \left(t - 1\right)} \right)}}{2}\right) e^{- p}}{p \sqrt{p \left(t - 1\right)}}
Simplificar
Respuesta [src] 
                                                                                                                               /                                                                                 _____             \                                  
                                                                                                                               |                                                                        ____    / -1       /  p   \|                                  
                                                                                                                               |                                                                      \/ pi *  /  --- *erfi|------||       //      1        for p = 0\
    /           _____ /                            /    ___   ___\       ___  -2*p \              _____     /    ___\\         |           _____ /      /    ___\       /    ___\          -p     \          \/    p       |  ____||       ||                        |
    |  ____    / -1   |      /    ___   ___\   erfi\I*\/ 2 *\/ p /   I*\/ 2 *e     |     ____    / -1       |p*\/ 2 ||  -p     |  ____    / -1   |  erfi\I*\/ p /   erfi\I*\/ p /       I*e       |                        \\/ -p /|  -p   || -2*p    -3*p           |
- 2*|\/ pi *  /  --- *|- erfi\I*\/ 2 *\/ p / + ------------------- - --------------| - \/ pi *  /  --- *erfi|-------||*e   + 2*|\/ pi *  /  --- *|- ------------- + ------------- - --------------| - -----------------------------|*e   + |
2(π−1p(−erfi⁡(ip)2+erfi⁡(ip)4p−ie−p2πp)−π−1perfi⁡(p−p)2)e−p−2(π−1p(−erfi⁡(2ip)+erfi⁡(2ip)4p−2ie−2p2πp)−π−1perfi⁡(2p−p))e−p+{1for p=0e−2pp−e−3ppotherwise2 \left(\sqrt{\pi} \sqrt{- \frac{1}{p}} \left(- \frac{\operatorname{erfi}{\left(i \sqrt{p} \right)}}{2} + \frac{\operatorname{erfi}{\left(i \sqrt{p} \right)}}{4 p} - \frac{i e^{- p}}{2 \sqrt{\pi} \sqrt{p}}\right) - \frac{\sqrt{\pi} \sqrt{- \frac{1}{p}} \operatorname{erfi}{\left(\frac{p}{\sqrt{- p}} \right)}}{2}\right) e^{- p} - 2 \left(\sqrt{\pi} \sqrt{- \frac{1}{p}} \left(- \operatorname{erfi}{\left(\sqrt{2} i \sqrt{p} \right)} + \frac{\operatorname{erfi}{\left(\sqrt{2} i \sqrt{p} \right)}}{4 p} - \frac{\sqrt{2} i e^{- 2 p}}{2 \sqrt{\pi} \sqrt{p}}\right) - \sqrt{\pi} \sqrt{- \frac{1}{p}} \operatorname{erfi}{\left(\frac{\sqrt{2} p}{\sqrt{- p}} \right)}\right) e^{- p} + \begin{cases} 1 & \text{for}\: p = 0 \\\frac{e^{- 2 p}}{p} - \frac{e^{- 3 p}}{p} & \text{otherwise} \end{cases} 
=
= 
                                                                                                                               /                                                                                 _____             \                                  
                                                                                                                               |                                                                        ____    / -1       /  p   \|                                  
                                                                                                                               |                                                                      \/ pi *  /  --- *erfi|------||       //      1        for p = 0\
    /           _____ /                            /    ___   ___\       ___  -2*p \              _____     /    ___\\         |           _____ /      /    ___\       /    ___\          -p     \          \/    p       |  ____||       ||                        |
    |  ____    / -1   |      /    ___   ___\   erfi\I*\/ 2 *\/ p /   I*\/ 2 *e     |     ____    / -1       |p*\/ 2 ||  -p     |  ____    / -1   |  erfi\I*\/ p /   erfi\I*\/ p /       I*e       |                        \\/ -p /|  -p   || -2*p    -3*p           |
- 2*|\/ pi *  /  --- *|- erfi\I*\/ 2 *\/ p / + ------------------- - --------------| - \/ pi *  /  --- *erfi|-------||*e   + 2*|\/ pi *  /  --- *|- ------------- + ------------- - --------------| - -----------------------------|*e   + |
2(π−1p(−erfi⁡(ip)2+erfi⁡(ip)4p−ie−p2πp)−π−1perfi⁡(p−p)2)e−p−2(π−1p(−erfi⁡(2ip)+erfi⁡(2ip)4p−2ie−2p2πp)−π−1perfi⁡(2p−p))e−p+{1for p=0e−2pp−e−3ppotherwise2 \left(\sqrt{\pi} \sqrt{- \frac{1}{p}} \left(- \frac{\operatorname{erfi}{\left(i \sqrt{p} \right)}}{2} + \frac{\operatorname{erfi}{\left(i \sqrt{p} \right)}}{4 p} - \frac{i e^{- p}}{2 \sqrt{\pi} \sqrt{p}}\right) - \frac{\sqrt{\pi} \sqrt{- \frac{1}{p}} \operatorname{erfi}{\left(\frac{p}{\sqrt{- p}} \right)}}{2}\right) e^{- p} - 2 \left(\sqrt{\pi} \sqrt{- \frac{1}{p}} \left(- \operatorname{erfi}{\left(\sqrt{2} i \sqrt{p} \right)} + \frac{\operatorname{erfi}{\left(\sqrt{2} i \sqrt{p} \right)}}{4 p} - \frac{\sqrt{2} i e^{- 2 p}}{2 \sqrt{\pi} \sqrt{p}}\right) - \sqrt{\pi} \sqrt{- \frac{1}{p}} \operatorname{erfi}{\left(\frac{\sqrt{2} p}{\sqrt{- p}} \right)}\right) e^{- p} + \begin{cases} 1 & \text{for}\: p = 0 \\\frac{e^{- 2 p}}{p} - \frac{e^{- 3 p}}{p} & \text{otherwise} \end{cases} 
-2*(sqrt(pi)*sqrt(-1/p)*(-erfi(i*sqrt(2)*sqrt(p)) + erfi(i*sqrt(2)*sqrt(p))/(4*p) - i*sqrt(2)*exp(-2*p)/(2*sqrt(pi)*sqrt(p))) - sqrt(pi)*sqrt(-1/p)*erfi(p*sqrt(2)/sqrt(-p)))*exp(-p) + 2*(sqrt(pi)*sqrt(-1/p)*(-erfi(i*sqrt(p))/2 + erfi(i*sqrt(p))/(4*p) - i*exp(-p)/(2*sqrt(pi)*sqrt(p))) - sqrt(pi)*sqrt(-1/p)*erfi(p/sqrt(-p))/2)*exp(-p) + Piecewise((1, p = 0), (exp(-2*p)/p - exp(-3*p)/p, True))

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

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