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