{
"query": {
"display": "$$\\int\\:t\\cdot\\:e^{t}dt$$",
"symbolab_question": "BIG_OPERATOR#\\int t\\cdot e^{t}dt"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Integrals",
"subTopic": "Indefinite Integrals",
"default": "e^{t}t-e^{t}+C",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$\\int\\:te^{t}dt=e^{t}t-e^{t}+C$$",
"input": "\\int\\:te^{t}dt",
"steps": [
{
"type": "interim",
"title": "Aplicar integración por partes",
"input": "\\int\\:te^{t}dt",
"steps": [
{
"type": "definition",
"title": "Definición de integración por partes",
"text": "$$\\int\\:uv'=uv-\\int\\:u'v$$"
},
{
"type": "step",
"primary": "$$u=t$$"
},
{
"type": "step",
"primary": "$$v'=e^{t}$$"
},
{
"type": "interim",
"title": "$$u'=\\frac{dt}{dt}=1$$",
"input": "\\frac{dt}{dt}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\frac{dt}{dt}=1$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuWV6zCUVy7FvtVpq63L1y1jqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIaSUX15NRU/Cc5dZXPCJmBB/+Ojk3WU3HnCJ1IoD4YQM"
}
},
{
"type": "interim",
"title": "$$v=\\int\\:e^{t}dt=e^{t}$$",
"input": "\\int\\:e^{t}dt",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de integración: $$\\int\\:e^{t}dt=e^{t}$$",
"result": "=e^{t}"
},
{
"type": "step",
"primary": "Agregar una constante a la solución",
"result": "=e^{t}+C",
"meta": {
"title": {
"extension": "Si $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ entonces $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$"
}
}
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/qhhaDZrZgUlETfur3sqot57+9tLAebyt/n2mi8F8LilkhUx2zhdhO4nW7WzpnEuQVYk1DIZNnsTKCEVZzifFGbdUnOYSjxn3jr9/Be/GRehMh0HBffmY/BeTvJSVkwFQ=="
}
},
{
"type": "step",
"result": "=te^{t}-\\int\\:1\\cdot\\:e^{t}dt"
},
{
"type": "interim",
"title": "Simplificar",
"input": "te^{t}-\\int\\:1\\cdot\\:e^{t}dt",
"result": "=e^{t}t-\\int\\:e^{t}dt",
"steps": [
{
"type": "step",
"primary": "Multiplicar: $$1\\cdot\\:e^{t}=e^{t}$$",
"result": "=e^{t}t-\\int\\:e^{t}dt"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Title 0Eq"
}
}
],
"meta": {
"interimType": "Integration By Parts 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73ws0Wor01EDiByq6fUP6OuFjYEyrvCWvj0UgozEWLeQeSQilmHh9Rr5YjxdBgITC8051+98xgjCy9hiyp0OchuC3J3pLlhawS5KF8FUXph+4WZeEfo9s4Fuv6tuGOH3zaqQhfqSx4Ip0S2H5wgufDKAmqfScVW3oCeVf4HD1fqfCflBNt8UaRAp4CrxZUS53b8yD3hLQ33B7/8/LpbPE3o="
}
},
{
"type": "step",
"result": "=e^{t}t-\\int\\:e^{t}dt"
},
{
"type": "interim",
"title": "$$\\int\\:e^{t}dt=e^{t}$$",
"input": "\\int\\:e^{t}dt",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de integración: $$\\int\\:e^{t}dt=e^{t}$$",
"result": "=e^{t}"
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/qhhaDZrZgUlETfur3sqot57+9tLAebyt/n2mi8F8LilkhUx2zhdhO4nW7WzpnEuclAbkiFkG93sSdf7uxW7VJJjpNCmRSmH3JH5GufgrBU4ZPxjZ4mnIgO8URhvMnutQ=="
}
},
{
"type": "step",
"result": "=e^{t}t-e^{t}"
},
{
"type": "step",
"primary": "Agregar una constante a la solución",
"result": "=e^{t}t-e^{t}+C",
"meta": {
"title": {
"extension": "Si $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ entonces $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$"
}
}
}
],
"meta": {
"solvingClass": "Integrals",
"practiceLink": "/practice/integration-practice#area=main&subtopic=Integration%20By%20Parts",
"practiceTopic": "Integration by Parts"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "t",
"plotRequest": "y=e^{t}t-e^{t}+C"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solución
Solución
Pasos de solución
Aplicar integración por partes
Agregar una constante a la solución