{
"query": {
"display": "$$\\int\\:\\frac{1}{x+1}dx$$",
"symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{x+1}dx"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Integrals",
"subTopic": "Indefinite Integrals",
"default": "\\ln\\left|x+1\\right|+C",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$\\int\\:\\frac{1}{x+1}dx=\\ln\\left|x+1\\right|+C$$",
"input": "\\int\\:\\frac{1}{x+1}dx",
"steps": [
{
"type": "interim",
"title": "Aplicar integración por sustitución",
"input": "\\int\\:\\frac{1}{x+1}dx",
"steps": [
{
"type": "definition",
"title": "Definición de integración por sustitución",
"text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$",
"secondary": [
"Sustituir: $$u=x+1$$"
]
},
{
"type": "interim",
"title": "$$\\frac{du}{dx}=1$$",
"input": "\\frac{d}{dx}\\left(x+1\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la suma/diferencia: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\frac{dx}{dx}+\\frac{d}{dx}\\left(1\\right)"
},
{
"type": "interim",
"title": "$$\\frac{dx}{dx}=1$$",
"input": "\\frac{dx}{dx}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\frac{dx}{dx}=1$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(1\\right)=0$$",
"input": "\\frac{d}{dx}\\left(1\\right)",
"steps": [
{
"type": "step",
"primary": "Derivada de una constante: $$\\frac{d}{dx}\\left({a}\\right)=0$$",
"result": "=0"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz"
}
},
{
"type": "step",
"result": "=1+0"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=1",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:du=1dx$$"
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:dx=1du$$"
},
{
"type": "step",
"result": "=\\int\\:\\frac{1}{u}\\cdot\\:1du"
},
{
"type": "step",
"result": "=\\int\\:\\frac{1}{u}du"
}
],
"meta": {
"interimType": "Integral U Substitution 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7xUNxvMK5+mw478WPt4Ez1YcjlLRK1jUV206qo4+vRN7yRnKSRJBHsnEXq1wS/zww7h8KUX9hN21wJE16H/NfZx2cv65xlj4FWO/jAv7Am1CptFvUOUfgDrM9m4ow9eu1Xql8XXPq6bNQlMm+36iNhmOWNNEJXCtWPvC8XGkEwD+5bXdImMl/a8RaF7ib5dZttgL93z3VYlmhU77lEbOh0U="
}
},
{
"type": "step",
"result": "=\\int\\:\\frac{1}{u}du"
},
{
"type": "step",
"primary": "Aplicar la regla de integración: $$\\int\\:\\frac{1}{u}du=\\ln\\left(\\left|u\\right|\\right)$$",
"result": "=\\ln\\left|u\\right|"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=x+1$$",
"result": "=\\ln\\left|x+1\\right|"
},
{
"type": "step",
"primary": "Agregar una constante a la solución",
"result": "=\\ln\\left|x+1\\right|+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=Substitution",
"practiceTopic": "Integral Substitution"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "y=\\ln\\left|x+1\\right|+C"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solución
Solución
Pasos de solución
Aplicar integración por sustitución
Aplicar la regla de integración:
Sustituir en la ecuación
Agregar una constante a la solución