Integral de a*exp(-a*x)*a*exp(-a*z*x)*abs(x) dx
Solución
Respuesta (Indefinida)
[src]
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| -a*x -a*z*x 2 | -a*x -a*z*x
| a*e *a*e *|x| dx = C + a * | |x|*e *e dx
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∫ a a e − a x e x − a z ∣ x ∣ d x = C + a 2 ∫ e − a x e x − a z ∣ x ∣ d x \int a a e^{- a x} e^{x - a z} \left|{x}\right|\, dx = C + a^{2} \int e^{- a x} e^{x - a z} \left|{x}\right|\, dx ∫ aa e − a x e x − a z ∣ x ∣ d x = C + a 2 ∫ e − a x e x − a z ∣ x ∣ d x
/ 1 / / pi pi\ / pi pi\ / pi pi\\
| -------- for Or|And||arg(a)| <= --, |arg(a) + arg(z)| < --|, And||arg(a) + arg(z)| <= --, |arg(a)| < --|, And||arg(a) + arg(z)| < --, |arg(a)| < --||
| 2 \ \ 2 2 / \ 2 2 / \ 2 2 //
| (1 + z)
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| oo
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| | 2 -a*x -a*x*z
| | a *|x|*e *e dx otherwise
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{ 1 ( z + 1 ) 2 for ( ∣ arg ( a ) ∣ ≤ π 2 ∧ ∣ arg ( a ) + arg ( z ) ∣ < π 2 ) ∨ ( ∣ arg ( a ) + arg ( z ) ∣ ≤ π 2 ∧ ∣ arg ( a ) ∣ < π 2 ) ∨ ( ∣ arg ( a ) + arg ( z ) ∣ < π 2 ∧ ∣ arg ( a ) ∣ < π 2 ) ∫ 0 ∞ a 2 e − a x e − a x z ∣ x ∣ d x otherwise \begin{cases} \frac{1}{\left(z + 1\right)^{2}} & \text{for}\: \left(\left|{\arg{\left(a \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(a \right)} + \arg{\left(z \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(a \right)} + \arg{\left(z \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(a \right)} + \arg{\left(z \right)}}\right| < \frac{\pi}{2} \wedge \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{0}^{\infty} a^{2} e^{- a x} e^{- a x z} \left|{x}\right|\, dx & \text{otherwise} \end{cases} ⎩ ⎨ ⎧ ( z + 1 ) 2 1 0 ∫ ∞ a 2 e − a x e − a x z ∣ x ∣ d x for ( ∣ arg ( a ) ∣ ≤ 2 π ∧ ∣ arg ( a ) + arg ( z ) ∣ < 2 π ) ∨ ( ∣ arg ( a ) + arg ( z ) ∣ ≤ 2 π ∧ ∣ arg ( a ) ∣ < 2 π ) ∨ ( ∣ arg ( a ) + arg ( z ) ∣ < 2 π ∧ ∣ arg ( a ) ∣ < 2 π ) otherwise
=
/ 1 / / pi pi\ / pi pi\ / pi pi\\
| -------- for Or|And||arg(a)| <= --, |arg(a) + arg(z)| < --|, And||arg(a) + arg(z)| <= --, |arg(a)| < --|, And||arg(a) + arg(z)| < --, |arg(a)| < --||
| 2 \ \ 2 2 / \ 2 2 / \ 2 2 //
| (1 + z)
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| oo
< /
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| | 2 -a*x -a*x*z
| | a *|x|*e *e dx otherwise
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\0
{ 1 ( z + 1 ) 2 for ( ∣ arg ( a ) ∣ ≤ π 2 ∧ ∣ arg ( a ) + arg ( z ) ∣ < π 2 ) ∨ ( ∣ arg ( a ) + arg ( z ) ∣ ≤ π 2 ∧ ∣ arg ( a ) ∣ < π 2 ) ∨ ( ∣ arg ( a ) + arg ( z ) ∣ < π 2 ∧ ∣ arg ( a ) ∣ < π 2 ) ∫ 0 ∞ a 2 e − a x e − a x z ∣ x ∣ d x otherwise \begin{cases} \frac{1}{\left(z + 1\right)^{2}} & \text{for}\: \left(\left|{\arg{\left(a \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(a \right)} + \arg{\left(z \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(a \right)} + \arg{\left(z \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(a \right)} + \arg{\left(z \right)}}\right| < \frac{\pi}{2} \wedge \left|{\arg{\left(a \right)}}\right| < \frac{\pi}{2}\right) \\\int\limits_{0}^{\infty} a^{2} e^{- a x} e^{- a x z} \left|{x}\right|\, dx & \text{otherwise} \end{cases} ⎩ ⎨ ⎧ ( z + 1 ) 2 1 0 ∫ ∞ a 2 e − a x e − a x z ∣ x ∣ d x for ( ∣ arg ( a ) ∣ ≤ 2 π ∧ ∣ arg ( a ) + arg ( z ) ∣ < 2 π ) ∨ ( ∣ arg ( a ) + arg ( z ) ∣ ≤ 2 π ∧ ∣ arg ( a ) ∣ < 2 π ) ∨ ( ∣ arg ( a ) + arg ( z ) ∣ < 2 π ∧ ∣ arg ( a ) ∣ < 2 π ) otherwise
Piecewise(((1 + z)^(-2), ((Abs(arg(a)) <= pi/2)∧(Abs(arg(a) + arg(z)) < pi/2))∨((Abs(arg(a)) < pi/2)∧(Abs(arg(a) + arg(z)) <= pi/2))∨((Abs(arg(a)) < pi/2)∧(Abs(arg(a) + arg(z)) < pi/2))), (Integral(a^2*|x|*exp(-a*x)*exp(-a*x*z), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.
