Sr Examen
Integral de (-sqrt(t-1)+1)*e^(-p*t) dt
Solución
Ha introducido
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3 / | | / _______ \ -p*t | \- \/ t - 1 + 1/*E dt | / 2
$$\int\limits_{2}^{3} e^{- p t} \left(1 - \sqrt{t - 1}\right)\, dt$$
Integral((-sqrt(t - 1) + 1)*E^((-p)*t), (t, 2, 3))
Respuesta (Indefinida)
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/ ____ / ____________\\
/ ________ | ____________ -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 |
\\ /
$$\int e^{- p t} \left(1 - \sqrt{t - 1}\right)\, dt = C + \begin{cases} t & \text{for}\: p = 0 \\- \frac{e^{- p t}}{p} & \text{otherwise} \end{cases} + \frac{\sqrt{t - 1} \left(\sqrt{p \left(t - 1\right)} e^{- p \left(t - 1\right)} + \frac{\sqrt{\pi} \operatorname{erfc}{\left(\sqrt{p \left(t - 1\right)} \right)}}{2}\right) e^{- p}}{p \sqrt{p \left(t - 1\right)}}$$
Respuesta
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/ _____ \
| ____ / -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 + |
$$2 \left(\sqrt{\pi} \sqrt{- \frac{1}{p}} \left(- \frac{\operatorname{erfi}{\left(i \sqrt{p} \right)}}{2} + \frac{\operatorname{erfi}{\left(i \sqrt{p} \right)}}{4 p} - \frac{i e^{- p}}{2 \sqrt{\pi} \sqrt{p}}\right) - \frac{\sqrt{\pi} \sqrt{- \frac{1}{p}} \operatorname{erfi}{\left(\frac{p}{\sqrt{- p}} \right)}}{2}\right) e^{- p} - 2 \left(\sqrt{\pi} \sqrt{- \frac{1}{p}} \left(- \operatorname{erfi}{\left(\sqrt{2} i \sqrt{p} \right)} + \frac{\operatorname{erfi}{\left(\sqrt{2} i \sqrt{p} \right)}}{4 p} - \frac{\sqrt{2} i e^{- 2 p}}{2 \sqrt{\pi} \sqrt{p}}\right) - \sqrt{\pi} \sqrt{- \frac{1}{p}} \operatorname{erfi}{\left(\frac{\sqrt{2} p}{\sqrt{- p}} \right)}\right) e^{- p} + \begin{cases} 1 & \text{for}\: p = 0 \\\frac{e^{- 2 p}}{p} - \frac{e^{- 3 p}}{p} & \text{otherwise} \end{cases}$$
=
=
/ _____ \
| ____ / -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 + |
$$2 \left(\sqrt{\pi} \sqrt{- \frac{1}{p}} \left(- \frac{\operatorname{erfi}{\left(i \sqrt{p} \right)}}{2} + \frac{\operatorname{erfi}{\left(i \sqrt{p} \right)}}{4 p} - \frac{i e^{- p}}{2 \sqrt{\pi} \sqrt{p}}\right) - \frac{\sqrt{\pi} \sqrt{- \frac{1}{p}} \operatorname{erfi}{\left(\frac{p}{\sqrt{- p}} \right)}}{2}\right) e^{- p} - 2 \left(\sqrt{\pi} \sqrt{- \frac{1}{p}} \left(- \operatorname{erfi}{\left(\sqrt{2} i \sqrt{p} \right)} + \frac{\operatorname{erfi}{\left(\sqrt{2} i \sqrt{p} \right)}}{4 p} - \frac{\sqrt{2} i e^{- 2 p}}{2 \sqrt{\pi} \sqrt{p}}\right) - \sqrt{\pi} \sqrt{- \frac{1}{p}} \operatorname{erfi}{\left(\frac{\sqrt{2} p}{\sqrt{- p}} \right)}\right) e^{- p} + \begin{cases} 1 & \text{for}\: p = 0 \\\frac{e^{- 2 p}}{p} - \frac{e^{- 3 p}}{p} & \text{otherwise} \end{cases}$$
-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.