Integral de exp(t)×sinat dx
Solución
Solución detallada
Usamos la integración por partes:
∫ u dv = u v − ∫ v du \int \operatorname{u} \operatorname{dv}
= \operatorname{u}\operatorname{v} -
\int \operatorname{v} \operatorname{du} ∫ u dv = u v − ∫ v du
que u ( t ) = sin ( a t ) u{\left(t \right)} = \sin{\left(a t \right)} u ( t ) = sin ( a t ) dv ( t ) = e t \operatorname{dv}{\left(t \right)} = e^{t} dv ( t ) = e t
Entonces du ( t ) = a cos ( a t ) \operatorname{du}{\left(t \right)} = a \cos{\left(a t \right)} du ( t ) = a cos ( a t )
Para buscar v ( t ) v{\left(t \right)} v ( t )
La integral de la función exponencial es la mesma.
∫ e t d t = e t \int e^{t}\, dt = e^{t} ∫ e t d t = e t
Ahora resolvemos podintegral.
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ a e t cos ( a t ) d t = a ∫ e t cos ( a t ) d t \int a e^{t} \cos{\left(a t \right)}\, dt = a \int e^{t} \cos{\left(a t \right)}\, dt ∫ a e t cos ( a t ) d t = a ∫ e t cos ( a t ) d t
Usamos la integración por partes:
∫ u dv = u v − ∫ v du \int \operatorname{u} \operatorname{dv}
= \operatorname{u}\operatorname{v} -
\int \operatorname{v} \operatorname{du} ∫ u dv = u v − ∫ v du
que u ( t ) = cos ( a t ) u{\left(t \right)} = \cos{\left(a t \right)} u ( t ) = cos ( a t ) dv ( t ) = e t \operatorname{dv}{\left(t \right)} = e^{t} dv ( t ) = e t
Entonces du ( t ) = − a sin ( a t ) \operatorname{du}{\left(t \right)} = - a \sin{\left(a t \right)} du ( t ) = − a sin ( a t )
Para buscar v ( t ) v{\left(t \right)} v ( t )
La integral de la función exponencial es la mesma.
∫ e t d t = e t \int e^{t}\, dt = e^{t} ∫ e t d t = e t
Ahora resolvemos podintegral.
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ ( − a e t sin ( a t ) ) d t = − a ∫ e t sin ( a t ) d t \int \left(- a e^{t} \sin{\left(a t \right)}\right)\, dt = - a \int e^{t} \sin{\left(a t \right)}\, dt ∫ ( − a e t sin ( a t ) ) d t = − a ∫ e t sin ( a t ) d t
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
{ − i ( t e t sinh ( t ) 2 − t e t cosh ( t ) 2 + e t cosh ( t ) 2 ) for a = − i i ( t e t sinh ( t ) 2 − t e t cosh ( t ) 2 + e t cosh ( t ) 2 ) for a = i − a e t cos ( a t ) a 2 + 1 + e t sin ( a t ) a 2 + 1 otherwese \begin{cases} - i \left(\frac{t e^{t} \sinh{\left(t \right)}}{2} - \frac{t e^{t} \cosh{\left(t \right)}}{2} + \frac{e^{t} \cosh{\left(t \right)}}{2}\right) & \text{for}\: a = - i \\i \left(\frac{t e^{t} \sinh{\left(t \right)}}{2} - \frac{t e^{t} \cosh{\left(t \right)}}{2} + \frac{e^{t} \cosh{\left(t \right)}}{2}\right) & \text{for}\: a = i \\- \frac{a e^{t} \cos{\left(a t \right)}}{a^{2} + 1} + \frac{e^{t} \sin{\left(a t \right)}}{a^{2} + 1} & \text{otherwese} \end{cases} ⎩ ⎨ ⎧ − i ( 2 t e t s i n h ( t ) − 2 t e t c o s h ( t ) + 2 e t c o s h ( t ) ) i ( 2 t e t s i n h ( t ) − 2 t e t c o s h ( t ) + 2 e t c o s h ( t ) ) − a 2 + 1 a e t c o s ( a t ) + a 2 + 1 e t s i n ( a t ) for a = − i for a = i otherwese
Por lo tanto, el resultado es: − a ( { − i ( t e t sinh ( t ) 2 − t e t cosh ( t ) 2 + e t cosh ( t ) 2 ) for a = − i i ( t e t sinh ( t ) 2 − t e t cosh ( t ) 2 + e t cosh ( t ) 2 ) for a = i − a e t cos ( a t ) a 2 + 1 + e t sin ( a t ) a 2 + 1 otherwese ) - a \left(\begin{cases} - i \left(\frac{t e^{t} \sinh{\left(t \right)}}{2} - \frac{t e^{t} \cosh{\left(t \right)}}{2} + \frac{e^{t} \cosh{\left(t \right)}}{2}\right) & \text{for}\: a = - i \\i \left(\frac{t e^{t} \sinh{\left(t \right)}}{2} - \frac{t e^{t} \cosh{\left(t \right)}}{2} + \frac{e^{t} \cosh{\left(t \right)}}{2}\right) & \text{for}\: a = i \\- \frac{a e^{t} \cos{\left(a t \right)}}{a^{2} + 1} + \frac{e^{t} \sin{\left(a t \right)}}{a^{2} + 1} & \text{otherwese} \end{cases}\right) − a ⎩ ⎨ ⎧ − i ( 2 t e t s i n h ( t ) − 2 t e t c o s h ( t ) + 2 e t c o s h ( t ) ) i ( 2 t e t s i n h ( t ) − 2 t e t c o s h ( t ) + 2 e t c o s h ( t ) ) − a 2 + 1 a e t c o s ( a t ) + a 2 + 1 e t s i n ( a t ) for a = − i for a = i otherwese
Por lo tanto, el resultado es: a ( a ( { − i ( t e t sinh ( t ) 2 − t e t cosh ( t ) 2 + e t cosh ( t ) 2 ) for a = − i i ( t e t sinh ( t ) 2 − t e t cosh ( t ) 2 + e t cosh ( t ) 2 ) for a = i − a e t cos ( a t ) a 2 + 1 + e t sin ( a t ) a 2 + 1 otherwese ) + e t cos ( a t ) ) a \left(a \left(\begin{cases} - i \left(\frac{t e^{t} \sinh{\left(t \right)}}{2} - \frac{t e^{t} \cosh{\left(t \right)}}{2} + \frac{e^{t} \cosh{\left(t \right)}}{2}\right) & \text{for}\: a = - i \\i \left(\frac{t e^{t} \sinh{\left(t \right)}}{2} - \frac{t e^{t} \cosh{\left(t \right)}}{2} + \frac{e^{t} \cosh{\left(t \right)}}{2}\right) & \text{for}\: a = i \\- \frac{a e^{t} \cos{\left(a t \right)}}{a^{2} + 1} + \frac{e^{t} \sin{\left(a t \right)}}{a^{2} + 1} & \text{otherwese} \end{cases}\right) + e^{t} \cos{\left(a t \right)}\right) a a ⎩ ⎨ ⎧ − i ( 2 t e t s i n h ( t ) − 2 t e t c o s h ( t ) + 2 e t c o s h ( t ) ) i ( 2 t e t s i n h ( t ) − 2 t e t c o s h ( t ) + 2 e t c o s h ( t ) ) − a 2 + 1 a e t c o s ( a t ) + a 2 + 1 e t s i n ( a t ) for a = − i for a = i otherwese + e t cos ( a t )
Ahora simplificar:
{ ( a ( i a ( t sinh ( t ) − t cosh ( t ) + cosh ( t ) ) − 2 cos ( a t ) ) 2 + sin ( a t ) ) e t for a = − i − ( a ( i a ( t sinh ( t ) − t cosh ( t ) + cosh ( t ) ) + 2 cos ( a t ) ) 2 − sin ( a t ) ) e t for a = i ( − a cos ( a t ) + sin ( a t ) ) e t a 2 + 1 otherwese \begin{cases} \left(\frac{a \left(i a \left(t \sinh{\left(t \right)} - t \cosh{\left(t \right)} + \cosh{\left(t \right)}\right) - 2 \cos{\left(a t \right)}\right)}{2} + \sin{\left(a t \right)}\right) e^{t} & \text{for}\: a = - i \\- \left(\frac{a \left(i a \left(t \sinh{\left(t \right)} - t \cosh{\left(t \right)} + \cosh{\left(t \right)}\right) + 2 \cos{\left(a t \right)}\right)}{2} - \sin{\left(a t \right)}\right) e^{t} & \text{for}\: a = i \\\frac{\left(- a \cos{\left(a t \right)} + \sin{\left(a t \right)}\right) e^{t}}{a^{2} + 1} & \text{otherwese} \end{cases} ⎩ ⎨ ⎧ ( 2 a ( ia ( t s i n h ( t ) − t c o s h ( t ) + c o s h ( t ) ) − 2 c o s ( a t ) ) + sin ( a t ) ) e t − ( 2 a ( ia ( t s i n h ( t ) − t c o s h ( t ) + c o s h ( t ) ) + 2 c o s ( a t ) ) − sin ( a t ) ) e t a 2 + 1 ( − a c o s ( a t ) + s i n ( a t ) ) e t for a = − i for a = i otherwese
Añadimos la constante de integración:
{ ( a ( i a ( t sinh ( t ) − t cosh ( t ) + cosh ( t ) ) − 2 cos ( a t ) ) 2 + sin ( a t ) ) e t for a = − i − ( a ( i a ( t sinh ( t ) − t cosh ( t ) + cosh ( t ) ) + 2 cos ( a t ) ) 2 − sin ( a t ) ) e t for a = i ( − a cos ( a t ) + sin ( a t ) ) e t a 2 + 1 otherwese + c o n s t a n t \begin{cases} \left(\frac{a \left(i a \left(t \sinh{\left(t \right)} - t \cosh{\left(t \right)} + \cosh{\left(t \right)}\right) - 2 \cos{\left(a t \right)}\right)}{2} + \sin{\left(a t \right)}\right) e^{t} & \text{for}\: a = - i \\- \left(\frac{a \left(i a \left(t \sinh{\left(t \right)} - t \cosh{\left(t \right)} + \cosh{\left(t \right)}\right) + 2 \cos{\left(a t \right)}\right)}{2} - \sin{\left(a t \right)}\right) e^{t} & \text{for}\: a = i \\\frac{\left(- a \cos{\left(a t \right)} + \sin{\left(a t \right)}\right) e^{t}}{a^{2} + 1} & \text{otherwese} \end{cases}+ \mathrm{constant} ⎩ ⎨ ⎧ ( 2 a ( ia ( t s i n h ( t ) − t c o s h ( t ) + c o s h ( t ) ) − 2 c o s ( a t ) ) + sin ( a t ) ) e t − ( 2 a ( ia ( t s i n h ( t ) − t c o s h ( t ) + c o s h ( t ) ) + 2 c o s ( a t ) ) − sin ( a t ) ) e t a 2 + 1 ( − a c o s ( a t ) + s i n ( a t ) ) e t for a = − i for a = i otherwese + constant
Respuesta:
{ ( a ( i a ( t sinh ( t ) − t cosh ( t ) + cosh ( t ) ) − 2 cos ( a t ) ) 2 + sin ( a t ) ) e t for a = − i − ( a ( i a ( t sinh ( t ) − t cosh ( t ) + cosh ( t ) ) + 2 cos ( a t ) ) 2 − sin ( a t ) ) e t for a = i ( − a cos ( a t ) + sin ( a t ) ) e t a 2 + 1 otherwese + c o n s t a n t \begin{cases} \left(\frac{a \left(i a \left(t \sinh{\left(t \right)} - t \cosh{\left(t \right)} + \cosh{\left(t \right)}\right) - 2 \cos{\left(a t \right)}\right)}{2} + \sin{\left(a t \right)}\right) e^{t} & \text{for}\: a = - i \\- \left(\frac{a \left(i a \left(t \sinh{\left(t \right)} - t \cosh{\left(t \right)} + \cosh{\left(t \right)}\right) + 2 \cos{\left(a t \right)}\right)}{2} - \sin{\left(a t \right)}\right) e^{t} & \text{for}\: a = i \\\frac{\left(- a \cos{\left(a t \right)} + \sin{\left(a t \right)}\right) e^{t}}{a^{2} + 1} & \text{otherwese} \end{cases}+ \mathrm{constant} ⎩ ⎨ ⎧ ( 2 a ( ia ( t s i n h ( t ) − t c o s h ( t ) + c o s h ( t ) ) − 2 c o s ( a t ) ) + sin ( a t ) ) e t − ( 2 a ( ia ( t s i n h ( t ) − t c o s h ( t ) + c o s h ( t ) ) + 2 c o s ( a t ) ) − sin ( a t ) ) e t a 2 + 1 ( − a c o s ( a t ) + s i n ( a t ) ) e t for a = − i for a = i otherwese + constant
Respuesta (Indefinida)
[src]
/ // / t t t\ \ \
| || |cosh(t)*e t*e *sinh(t) t*cosh(t)*e | | |
| ||-I*|---------- + ------------ - ------------| for a = -I| |
| || \ 2 2 2 / | |
/ | || | |
| | || / t t t\ | |
| t t | || |cosh(t)*e t*e *sinh(t) t*cosh(t)*e | | t|
| e *sin(a*t) dt = C + e *sin(a*t) - a*|a*|
∫ e t sin ( a t ) d t = C − a ( a ( { − i ( t e t sinh ( t ) 2 − t e t cosh ( t ) 2 + e t cosh ( t ) 2 ) for a = − i i ( t e t sinh ( t ) 2 − t e t cosh ( t ) 2 + e t cosh ( t ) 2 ) for a = i − a e t cos ( a t ) a 2 + 1 + e t sin ( a t ) a 2 + 1 otherwise ) + e t cos ( a t ) ) + e t sin ( a t ) \int e^{t} \sin{\left(a t \right)}\, dt = C - a \left(a \left(\begin{cases} - i \left(\frac{t e^{t} \sinh{\left(t \right)}}{2} - \frac{t e^{t} \cosh{\left(t \right)}}{2} + \frac{e^{t} \cosh{\left(t \right)}}{2}\right) & \text{for}\: a = - i \\i \left(\frac{t e^{t} \sinh{\left(t \right)}}{2} - \frac{t e^{t} \cosh{\left(t \right)}}{2} + \frac{e^{t} \cosh{\left(t \right)}}{2}\right) & \text{for}\: a = i \\- \frac{a e^{t} \cos{\left(a t \right)}}{a^{2} + 1} + \frac{e^{t} \sin{\left(a t \right)}}{a^{2} + 1} & \text{otherwise} \end{cases}\right) + e^{t} \cos{\left(a t \right)}\right) + e^{t} \sin{\left(a t \right)} ∫ e t sin ( a t ) d t = C − a a ⎩ ⎨ ⎧ − i ( 2 t e t s i n h ( t ) − 2 t e t c o s h ( t ) + 2 e t c o s h ( t ) ) i ( 2 t e t s i n h ( t ) − 2 t e t c o s h ( t ) + 2 e t c o s h ( t ) ) − a 2 + 1 a e t c o s ( a t ) + a 2 + 1 e t s i n ( a t ) for a = − i for a = i otherwise + e t cos ( a t ) + e t sin ( a t )
a E*sin(a) E*a*cos(a)
------ + -------- - ----------
2 2 2
1 + a 1 + a 1 + a
− e a cos ( a ) a 2 + 1 + a a 2 + 1 + e sin ( a ) a 2 + 1 - \frac{e a \cos{\left(a \right)}}{a^{2} + 1} + \frac{a}{a^{2} + 1} + \frac{e \sin{\left(a \right)}}{a^{2} + 1} − a 2 + 1 e a cos ( a ) + a 2 + 1 a + a 2 + 1 e sin ( a )
=
a E*sin(a) E*a*cos(a)
------ + -------- - ----------
2 2 2
1 + a 1 + a 1 + a
− e a cos ( a ) a 2 + 1 + a a 2 + 1 + e sin ( a ) a 2 + 1 - \frac{e a \cos{\left(a \right)}}{a^{2} + 1} + \frac{a}{a^{2} + 1} + \frac{e \sin{\left(a \right)}}{a^{2} + 1} − a 2 + 1 e a cos ( a ) + a 2 + 1 a + a 2 + 1 e sin ( a )
a/(1 + a^2) + E*sin(a)/(1 + a^2) - E*a*cos(a)/(1 + a^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.
