{
"query": {
"display": "integral $$\\cos^{2}\\left(x\\right)$$",
"symbolab_question": "PRE_CALC#integral \\cos^{2}(x)"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Integrals",
"subTopic": "Indefinite Integrals",
"default": "\\frac{1}{2}(x+\\frac{1}{2}\\sin(2x))+C",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$\\int\\:\\cos^{2}\\left(x\\right)dx=\\frac{1}{2}\\left(x+\\frac{1}{2}\\sin\\left(2x\\right)\\right)+C$$",
"input": "\\int\\:\\cos^{2}\\left(x\\right)dx",
"steps": [
{
"type": "interim",
"title": "Re-escribir usando identidades trigonométricas",
"input": "\\int\\:\\cos^{2}\\left(x\\right)dx",
"result": "=\\int\\:\\frac{1+\\cos\\left(2x\\right)}{2}dx",
"steps": [
{
"type": "step",
"primary": "Usar la siguiente identidad: $$\\cos^{2}\\left(x\\right)=\\frac{1+\\cos\\left(2x\\right)}{2}$$",
"result": "=\\int\\:\\frac{1+\\cos\\left(2x\\right)}{2}dx"
}
],
"meta": {
"interimType": "Trig Rewrite Using Trig identities 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7zmxp6EylfVUAAN+93J/vCP1fWutlCU4zUzgkFchUVlKOeWgsE4Mk40prEDZkKQ8Xcq64+b8YguXf4qCtKW9b7JFhxSzNcjgPzX10I7nnzbP3KDBHeUGKQrBZNh7BVFgRiWQYJBNrvQ2kgC5s4TyrECBBTEk/JQ2cZ9WKuRzClU7k3Tbt2XGVutgJtuOwcqSe/6qvCrlX5Dy5AkBbQ6+UTsbvAE3NEPNsr/I9UpryYZdxQDnWhmZm3E9/bRYH1Xn0A=="
}
},
{
"type": "step",
"primary": "Sacar la constante: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=\\frac{1}{2}\\cdot\\:\\int\\:1+\\cos\\left(2x\\right)dx"
},
{
"type": "step",
"primary": "Aplicar la regla de la suma: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$",
"result": "=\\frac{1}{2}\\left(\\int\\:1dx+\\int\\:\\cos\\left(2x\\right)dx\\right)"
},
{
"type": "interim",
"title": "$$\\int\\:1dx=x$$",
"input": "\\int\\:1dx",
"steps": [
{
"type": "step",
"primary": "Integral de una constante: $$\\int{a}dx=ax$$",
"result": "=1\\cdot\\:x"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=x",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "interim",
"title": "$$\\int\\:\\cos\\left(2x\\right)dx=\\frac{1}{2}\\sin\\left(2x\\right)$$",
"input": "\\int\\:\\cos\\left(2x\\right)dx",
"steps": [
{
"type": "interim",
"title": "Aplicar integración por sustitución",
"input": "\\int\\:\\cos\\left(2x\\right)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: $$u=2x$$"
]
},
{
"type": "interim",
"title": "$$\\frac{du}{dx}=2$$",
"input": "\\frac{d}{dx}\\left(2x\\right)",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=2\\frac{dx}{dx}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\frac{dx}{dx}=1$$",
"result": "=2\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=2",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51iZ4v02Fm2dNqQJZnxCX4Je"
}
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:du=2dx$$"
},
{
"type": "step",
"primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{2}du$$"
},
{
"type": "step",
"result": "=\\int\\:\\cos\\left(u\\right)\\frac{1}{2}du"
}
],
"meta": {
"interimType": "Integral U Substitution 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74q6JkKmzW6HevVc80KBrFssjvX7KVUO/AeCFSId4S33cZqb2ujmN2FEZC5M/msYIHiX35dQ/h01lIvxamZtt5Pfzl9vMjUAUvB5H3kSKKYo4Ruz4fIbA7DSu1jg0w5EAYEFMST8lDZxn1Yq5HMKVTt7TijpDXoC59St8qKhyOfy7KGJ5+URaFG9aClfqFsGkYz4YZzEQj4DuiGtgcX4ib0="
}
},
{
"type": "step",
"result": "=\\int\\:\\cos\\left(u\\right)\\frac{1}{2}du"
},
{
"type": "step",
"primary": "Sacar la constante: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=\\frac{1}{2}\\cdot\\:\\int\\:\\cos\\left(u\\right)du"
},
{
"type": "step",
"primary": "Aplicar la regla de integración: $$\\int\\:\\cos\\left(u\\right)du=\\sin\\left(u\\right)$$",
"result": "=\\frac{1}{2}\\sin\\left(u\\right)"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=2x$$",
"result": "=\\frac{1}{2}\\sin\\left(2x\\right)"
}
],
"meta": {
"solvingClass": "Integrals",
"interimType": "Integrals"
}
},
{
"type": "step",
"result": "=\\frac{1}{2}\\left(x+\\frac{1}{2}\\sin\\left(2x\\right)\\right)"
},
{
"type": "step",
"primary": "Agregar una constante a la solución",
"result": "=\\frac{1}{2}\\left(x+\\frac{1}{2}\\sin\\left(2x\\right)\\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",
"practiceTopic": "Integrals"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "y=\\frac{1}{2}(x+\\frac{1}{2}\\sin(2x))+C"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solución
integral
Solución
Pasos de solución
Re-escribir usando identidades trigonométricas
Sacar la constante:
Aplicar la regla de la suma:
Agregar una constante a la solución