Sr Examen
Integral de (1+a*x)*exp(-a*x)*cos(w*x) dx
Solución
Ha introducido
[src]
1 / | | -a*x | (1 + a*x)*e *cos(w*x) dx | / 0
$$\int\limits_{0}^{1} \left(a x + 1\right) e^{- a x} \cos{\left(w x \right)}\, dx$$
Integral(((1 + a*x)*exp((-a)*x))*cos(w*x), (x, 0, 1))
Respuesta
[src]
/ 1 for Or(And(a = 0, w = 0), And(a = 0, a = -I*w, w = 0), And(a = 0, a = I*w, w = 0), And(a = 0, a = -I*w, a = I*w, w = 0)) | | I*w I*w I*w I*w I*w | cos(w)*e 3*I 3*I*e *sin(w) w*e *sin(w) 3*I*cos(w)*e I*w*cos(w)*e | ----------- + --- - --------------- - ------------- - --------------- - --------------- for Or(And(a = 0, a = -I*w), And(a = -I*w, a = I*w), And(a = -I*w, w = 0), And(a = 0, a = -I*w, a = I*w), And(a = -I*w, a = I*w, w = 0), a = -I*w) | 4 4*w 4 4 4*w 4 | | -I*w -I*w -I*w -I*w -I*w < 3*I cos(w)*e w*e *sin(w) 3*I*e *sin(w) I*w*cos(w)*e 3*I*cos(w)*e | - --- + ------------ - -------------- + ---------------- + ---------------- + ---------------- for Or(And(a = 0, a = I*w), And(a = I*w, w = 0), a = I*w) | 4*w 4 4 4 4 4*w | | 3 3 4 3 3 3 2 2 2 | 2*a w *sin(w) a *cos(w) 2*a *cos(w) a*w *sin(w) w*a *sin(w) a *w *cos(w) 3*w*a *sin(w) |----------------- + -------------------------- - -------------------------- - -------------------------- + -------------------------- + -------------------------- - -------------------------- + -------------------------- otherwise | 4 4 2 2 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a \a + w + 2*a *w a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e
$$\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge a = i w \wedge w = 0\right) \\- \frac{w e^{i w} \sin{\left(w \right)}}{4} - \frac{i w e^{i w} \cos{\left(w \right)}}{4} - \frac{3 i e^{i w} \sin{\left(w \right)}}{4} + \frac{e^{i w} \cos{\left(w \right)}}{4} - \frac{3 i e^{i w} \cos{\left(w \right)}}{4 w} + \frac{3 i}{4 w} & \text{for}\: \left(a = 0 \wedge a = - i w\right) \vee \left(a = - i w \wedge a = i w\right) \vee \left(a = - i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge a = i w\right) \vee \left(a = - i w \wedge a = i w \wedge w = 0\right) \vee a = - i w \\- \frac{w e^{- i w} \sin{\left(w \right)}}{4} + \frac{i w e^{- i w} \cos{\left(w \right)}}{4} + \frac{3 i e^{- i w} \sin{\left(w \right)}}{4} + \frac{e^{- i w} \cos{\left(w \right)}}{4} - \frac{3 i}{4 w} + \frac{3 i e^{- i w} \cos{\left(w \right)}}{4 w} & \text{for}\: \left(a = 0 \wedge a = i w\right) \vee \left(a = i w \wedge w = 0\right) \vee a = i w \\- \frac{a^{4} \cos{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{a^{3} w \sin{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} - \frac{2 a^{3} \cos{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{2 a^{3}}{a^{4} + 2 a^{2} w^{2} + w^{4}} - \frac{a^{2} w^{2} \cos{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{3 a^{2} w \sin{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{a w^{3} \sin{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{w^{3} \sin{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} & \text{otherwise} \end{cases}$$
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/ 1 for Or(And(a = 0, w = 0), And(a = 0, a = -I*w, w = 0), And(a = 0, a = I*w, w = 0), And(a = 0, a = -I*w, a = I*w, w = 0)) | | I*w I*w I*w I*w I*w | cos(w)*e 3*I 3*I*e *sin(w) w*e *sin(w) 3*I*cos(w)*e I*w*cos(w)*e | ----------- + --- - --------------- - ------------- - --------------- - --------------- for Or(And(a = 0, a = -I*w), And(a = -I*w, a = I*w), And(a = -I*w, w = 0), And(a = 0, a = -I*w, a = I*w), And(a = -I*w, a = I*w, w = 0), a = -I*w) | 4 4*w 4 4 4*w 4 | | -I*w -I*w -I*w -I*w -I*w < 3*I cos(w)*e w*e *sin(w) 3*I*e *sin(w) I*w*cos(w)*e 3*I*cos(w)*e | - --- + ------------ - -------------- + ---------------- + ---------------- + ---------------- for Or(And(a = 0, a = I*w), And(a = I*w, w = 0), a = I*w) | 4*w 4 4 4 4 4*w | | 3 3 4 3 3 3 2 2 2 | 2*a w *sin(w) a *cos(w) 2*a *cos(w) a*w *sin(w) w*a *sin(w) a *w *cos(w) 3*w*a *sin(w) |----------------- + -------------------------- - -------------------------- - -------------------------- + -------------------------- + -------------------------- - -------------------------- + -------------------------- otherwise | 4 4 2 2 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a 4 a 4 a 2 2 a \a + w + 2*a *w a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e a *e + w *e + 2*a *w *e
$$\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge a = i w \wedge w = 0\right) \\- \frac{w e^{i w} \sin{\left(w \right)}}{4} - \frac{i w e^{i w} \cos{\left(w \right)}}{4} - \frac{3 i e^{i w} \sin{\left(w \right)}}{4} + \frac{e^{i w} \cos{\left(w \right)}}{4} - \frac{3 i e^{i w} \cos{\left(w \right)}}{4 w} + \frac{3 i}{4 w} & \text{for}\: \left(a = 0 \wedge a = - i w\right) \vee \left(a = - i w \wedge a = i w\right) \vee \left(a = - i w \wedge w = 0\right) \vee \left(a = 0 \wedge a = - i w \wedge a = i w\right) \vee \left(a = - i w \wedge a = i w \wedge w = 0\right) \vee a = - i w \\- \frac{w e^{- i w} \sin{\left(w \right)}}{4} + \frac{i w e^{- i w} \cos{\left(w \right)}}{4} + \frac{3 i e^{- i w} \sin{\left(w \right)}}{4} + \frac{e^{- i w} \cos{\left(w \right)}}{4} - \frac{3 i}{4 w} + \frac{3 i e^{- i w} \cos{\left(w \right)}}{4 w} & \text{for}\: \left(a = 0 \wedge a = i w\right) \vee \left(a = i w \wedge w = 0\right) \vee a = i w \\- \frac{a^{4} \cos{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{a^{3} w \sin{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} - \frac{2 a^{3} \cos{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{2 a^{3}}{a^{4} + 2 a^{2} w^{2} + w^{4}} - \frac{a^{2} w^{2} \cos{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{3 a^{2} w \sin{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{a w^{3} \sin{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} + \frac{w^{3} \sin{\left(w \right)}}{a^{4} e^{a} + 2 a^{2} w^{2} e^{a} + w^{4} e^{a}} & \text{otherwise} \end{cases}$$
Piecewise((1, ((a = 0)∧(w = 0))∨((a = 0)∧(w = 0)∧(a = i*w))∨((a = 0)∧(w = 0)∧(a = -i*w))∨((a = 0)∧(w = 0)∧(a = i*w)∧(a = -i*w))), (cos(w)*exp(i*w)/4 + 3*i/(4*w) - 3*i*exp(i*w)*sin(w)/4 - w*exp(i*w)*sin(w)/4 - 3*i*cos(w)*exp(i*w)/(4*w) - i*w*cos(w)*exp(i*w)/4, (a = -i*w)∨((a = 0)∧(a = -i*w))∨((w = 0)∧(a = -i*w))∨((a = i*w)∧(a = -i*w))∨((a = 0)∧(a = i*w)∧(a = -i*w))∨((w = 0)∧(a = i*w)∧(a = -i*w))), (-3*i/(4*w) + cos(w)*exp(-i*w)/4 - w*exp(-i*w)*sin(w)/4 + 3*i*exp(-i*w)*sin(w)/4 + i*w*cos(w)*exp(-i*w)/4 + 3*i*cos(w)*exp(-i*w)/(4*w), (a = i*w)∨((a = 0)∧(a = i*w))∨((w = 0)∧(a = i*w))), (2*a^3/(a^4 + w^4 + 2*a^2*w^2) + w^3*sin(w)/(a^4*exp(a) + w^4*exp(a) + 2*a^2*w^2*exp(a)) - a^4*cos(w)/(a^4*exp(a) + w^4*exp(a) + 2*a^2*w^2*exp(a)) - 2*a^3*cos(w)/(a^4*exp(a) + w^4*exp(a) + 2*a^2*w^2*exp(a)) + a*w^3*sin(w)/(a^4*exp(a) + w^4*exp(a) + 2*a^2*w^2*exp(a)) + w*a^3*sin(w)/(a^4*exp(a) + w^4*exp(a) + 2*a^2*w^2*exp(a)) - a^2*w^2*cos(w)/(a^4*exp(a) + w^4*exp(a) + 2*a^2*w^2*exp(a)) + 3*w*a^2*sin(w)/(a^4*exp(a) + w^4*exp(a) + 2*a^2*w^2*exp(a)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.