Sr Examen
Integral de e^(ax)sinmxdx dx
Solución
Ha introducido
[src]
1 / | | a*x | E *sin(m*x) dx | / 0
$$\int\limits_{0}^{1} e^{a x} \sin{\left(m x \right)}\, dx$$
Integral(E^(a*x)*sin(m*x), (x, 0, 1))
Respuesta (Indefinida)
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// 0 for And(a = 0, m = 0)\
|| |
|| -I*m*x -I*m*x -I*m*x |
||x*e *sin(m*x) I*e *sin(m*x) I*x*cos(m*x)*e |
||------------------ + ------------------ - -------------------- for a = -I*m |
/ || 2 2*m 2 |
| || |
| a*x || I*m*x I*m*x I*m*x |
| E *sin(m*x) dx = C + |< x*e *sin(m*x) I*x*cos(m*x)*e I*e *sin(m*x) |
| || ----------------- + ------------------- - ----------------- for a = I*m |
/ || 2 2 2*m |
|| |
|| a*x a*x |
|| a*e *sin(m*x) m*cos(m*x)*e |
|| --------------- - --------------- otherwise |
|| 2 2 2 2 |
\\ a + m a + m /
$$\int e^{a x} \sin{\left(m x \right)}\, dx = C + \begin{cases} 0 & \text{for}\: a = 0 \wedge m = 0 \\\frac{x e^{- i m x} \sin{\left(m x \right)}}{2} - \frac{i x e^{- i m x} \cos{\left(m x \right)}}{2} + \frac{i e^{- i m x} \sin{\left(m x \right)}}{2 m} & \text{for}\: a = - i m \\\frac{x e^{i m x} \sin{\left(m x \right)}}{2} + \frac{i x e^{i m x} \cos{\left(m x \right)}}{2} - \frac{i e^{i m x} \sin{\left(m x \right)}}{2 m} & \text{for}\: a = i m \\\frac{a e^{a x} \sin{\left(m x \right)}}{a^{2} + m^{2}} - \frac{m e^{a x} \cos{\left(m x \right)}}{a^{2} + m^{2}} & \text{otherwise} \end{cases}$$
Respuesta
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/ 0 for Or(And(a = 0, m = 0), And(a = 0, a = -I*m, m = 0), And(a = 0, a = I*m, m = 0), And(a = 0, a = -I*m, a = I*m, m = 0)) | | -I*m -I*m -I*m | 1 e *sin(m) I*cos(m)*e cos(m)*e |--- + ------------ - -------------- - ------------ for Or(And(a = 0, a = -I*m), And(a = -I*m, a = I*m), And(a = -I*m, m = 0), And(a = 0, a = -I*m, a = I*m), And(a = -I*m, a = I*m, m = 0), a = -I*m) |2*m 2 2 2*m | | I*m I*m I*m < 1 e *sin(m) I*cos(m)*e cos(m)*e | --- + ----------- + ------------- - ----------- for Or(And(a = 0, a = I*m), And(a = I*m, m = 0), a = I*m) | 2*m 2 2 2*m | | a a | m a*e *sin(m) m*cos(m)*e | ------- + ----------- - ----------- otherwise | 2 2 2 2 2 2 \ a + m a + m a + m
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge a = i m \wedge m = 0\right) \\\frac{e^{- i m} \sin{\left(m \right)}}{2} - \frac{i e^{- i m} \cos{\left(m \right)}}{2} + \frac{1}{2 m} - \frac{e^{- i m} \cos{\left(m \right)}}{2 m} & \text{for}\: \left(a = 0 \wedge a = - i m\right) \vee \left(a = - i m \wedge a = i m\right) \vee \left(a = - i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge a = i m\right) \vee \left(a = - i m \wedge a = i m \wedge m = 0\right) \vee a = - i m \\\frac{e^{i m} \sin{\left(m \right)}}{2} + \frac{i e^{i m} \cos{\left(m \right)}}{2} - \frac{e^{i m} \cos{\left(m \right)}}{2 m} + \frac{1}{2 m} & \text{for}\: \left(a = 0 \wedge a = i m\right) \vee \left(a = i m \wedge m = 0\right) \vee a = i m \\\frac{a e^{a} \sin{\left(m \right)}}{a^{2} + m^{2}} - \frac{m e^{a} \cos{\left(m \right)}}{a^{2} + m^{2}} + \frac{m}{a^{2} + m^{2}} & \text{otherwise} \end{cases}$$
=
=
/ 0 for Or(And(a = 0, m = 0), And(a = 0, a = -I*m, m = 0), And(a = 0, a = I*m, m = 0), And(a = 0, a = -I*m, a = I*m, m = 0)) | | -I*m -I*m -I*m | 1 e *sin(m) I*cos(m)*e cos(m)*e |--- + ------------ - -------------- - ------------ for Or(And(a = 0, a = -I*m), And(a = -I*m, a = I*m), And(a = -I*m, m = 0), And(a = 0, a = -I*m, a = I*m), And(a = -I*m, a = I*m, m = 0), a = -I*m) |2*m 2 2 2*m | | I*m I*m I*m < 1 e *sin(m) I*cos(m)*e cos(m)*e | --- + ----------- + ------------- - ----------- for Or(And(a = 0, a = I*m), And(a = I*m, m = 0), a = I*m) | 2*m 2 2 2*m | | a a | m a*e *sin(m) m*cos(m)*e | ------- + ----------- - ----------- otherwise | 2 2 2 2 2 2 \ a + m a + m a + m
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge a = i m \wedge m = 0\right) \\\frac{e^{- i m} \sin{\left(m \right)}}{2} - \frac{i e^{- i m} \cos{\left(m \right)}}{2} + \frac{1}{2 m} - \frac{e^{- i m} \cos{\left(m \right)}}{2 m} & \text{for}\: \left(a = 0 \wedge a = - i m\right) \vee \left(a = - i m \wedge a = i m\right) \vee \left(a = - i m \wedge m = 0\right) \vee \left(a = 0 \wedge a = - i m \wedge a = i m\right) \vee \left(a = - i m \wedge a = i m \wedge m = 0\right) \vee a = - i m \\\frac{e^{i m} \sin{\left(m \right)}}{2} + \frac{i e^{i m} \cos{\left(m \right)}}{2} - \frac{e^{i m} \cos{\left(m \right)}}{2 m} + \frac{1}{2 m} & \text{for}\: \left(a = 0 \wedge a = i m\right) \vee \left(a = i m \wedge m = 0\right) \vee a = i m \\\frac{a e^{a} \sin{\left(m \right)}}{a^{2} + m^{2}} - \frac{m e^{a} \cos{\left(m \right)}}{a^{2} + m^{2}} + \frac{m}{a^{2} + m^{2}} & \text{otherwise} \end{cases}$$
Piecewise((0, ((a = 0)∧(m = 0))∨((a = 0)∧(m = 0)∧(a = i*m))∨((a = 0)∧(m = 0)∧(a = -i*m))∨((a = 0)∧(m = 0)∧(a = i*m)∧(a = -i*m))), (1/(2*m) + exp(-i*m)*sin(m)/2 - i*cos(m)*exp(-i*m)/2 - cos(m)*exp(-i*m)/(2*m), (a = -i*m)∨((a = 0)∧(a = -i*m))∨((m = 0)∧(a = -i*m))∨((a = i*m)∧(a = -i*m))∨((a = 0)∧(a = i*m)∧(a = -i*m))∨((m = 0)∧(a = i*m)∧(a = -i*m))), (1/(2*m) + exp(i*m)*sin(m)/2 + i*cos(m)*exp(i*m)/2 - cos(m)*exp(i*m)/(2*m), (a = i*m)∨((a = 0)∧(a = i*m))∨((m = 0)∧(a = i*m))), (m/(a^2 + m^2) + a*exp(a)*sin(m)/(a^2 + m^2) - m*cos(m)*exp(a)/(a^2 + m^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.