integral de sec(sqrt(x+1))tan(sqrt(x+1))

{ "query": { "display": "$$\\int\\:\\sec\\left(\\sqrt{x+1}\\right)\\tan\\left(\\sqrt{x+1}\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\sec(\\sqrt{x+1})\\tan(\\sqrt{x+1})dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "2(\\sqrt{x+1}\\sec(\\sqrt{x+1})-\\ln\\left|\\tan(\\sqrt{x+1})+\\sec(\\sqrt{x+1})\\right|)+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\sec\\left(\\sqrt{x+1}\\right)\\tan\\left(\\sqrt{x+1}\\right)dx=2\\left(\\sqrt{x+1}\\sec\\left(\\sqrt{x+1}\\right)-\\ln\\left|\\tan\\left(\\sqrt{x+1}\\right)+\\sec\\left(\\sqrt{x+1}\\right)\\right|\\right)+C$$", "input": "\\int\\:\\sec\\left(\\sqrt{x+1}\\right)\\tan\\left(\\sqrt{x+1}\\right)dx", "steps": [ { "type": "interim", "title": "Aplicar integración por sustitución", "input": "\\int\\:\\sec\\left(\\sqrt{x+1}\\right)\\tan\\left(\\sqrt{x+1}\\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: $$u=x+1$$" ] }, { "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\\:\\sec\\left(\\sqrt{u}\\right)\\tan\\left(\\sqrt{u}\\right)\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:\\sec\\left(\\sqrt{u}\\right)\\tan\\left(\\sqrt{u}\\right)du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s70wYmVHHT0YkMfQPerN/+OZDvSqlNlVBCti4gW0XfjtjQpz0jiKJSwn07qQxumvgCnEjypn9AjSlgvJBiJApN19yrccDNZ+6MHaBthhc+pLHbHy9hmFUb8831ZbMI8TtmMszQOKHqrqVlcLQhS3yJz+HVKIb7bDhrg4iruP46eM+qBmO88/44frwpjBYq5/ITgeLly0fL3FNLbSRyi8EKC5LUjnKU9BAiUeAaKPy/ftiGNe8E5bqGoINpWJaKZx67YmpXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "=\\int\\:\\sec\\left(\\sqrt{u}\\right)\\tan\\left(\\sqrt{u}\\right)du" }, { "type": "interim", "title": "Aplicar integración por sustitución", "input": "\\int\\:\\sec\\left(\\sqrt{u}\\right)\\tan\\left(\\sqrt{u}\\right)du", "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: $$v=\\sqrt{u}$$" ] }, { "type": "interim", "title": "$$\\frac{dv}{du}=\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{d}{du}\\left(\\sqrt{u}\\right)", "steps": [ { "type": "step", "primary": "Aplicar las leyes de los exponentes: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{du}\\left(u^{\\frac{1}{2}}\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Aplicar la regla de la potencia: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{1}{2}u^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplificar $$\\frac{1}{2}u^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{1}{2}u^{\\frac{1}{2}-1}", "result": "=\\frac{1}{2\\sqrt{u}}", "steps": [ { "type": "interim", "title": "$$u^{\\frac{1}{2}-1}=u^{-\\frac{1}{2}}$$", "input": "u^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Simplificar $$\\frac{1}{2}-1\\:$$en una fracción:$${\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=u^{-\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convertir a fracción: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1\\cdot\\:2}{2}+\\frac{1}{2}" }, { "type": "step", "primary": "Ya que los denominadores son iguales, combinar las fracciones: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$-1\\cdot\\:2+1=-1$$", "input": "-1\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiplicar los numeros: $$1\\cdot\\:2=2$$", "result": "=-2+1" }, { "type": "step", "primary": "Sumar/restar lo siguiente: $$-2+1=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s731snK5z/nd3Sq/6JpCqiX1XTSum/z5kLpMzXS1UJIew02FKSBoQo9V3G05AlnWtTyCE30rzMlUAIVDyhseMBropKGn5MuXZnb2ZCo/hVsBU=" } }, { "type": "step", "result": "=\\frac{-1}{2}" }, { "type": "step", "primary": "Aplicar las propiedades de las fracciones: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VcI2MpaClJgyGWg1EkySKe0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjMvOxDqXzE3/CFO0TFmffHAH2kDe5DGYTz3TrPquGdIhyukSOA/1RgMKO0TMhInPOMabdUggEogUL9RT7PNKh0VQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}u^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Aplicar las leyes de los exponentes: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{u}}$$" ], "result": "=\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{u}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiplicar fracciones: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\sqrt{u}}" }, { "type": "step", "primary": "Multiplicar los numeros: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{u}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JOPQ2g2GS9EQptV8nckZSrH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJv2RkT96g5Q5jVbn5fyeQzwB9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC0fckB6tn+Lslm20bk1NfC+RFXBEVoMC309dBjB0EbJM4Es6agjDQJIYZAr5O37aAQ==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=\\frac{1}{2\\sqrt{u}}du$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2\\sqrt{u}dv$$" }, { "type": "step", "result": "=\\int\\:\\sec\\left(v\\right)\\tan\\left(v\\right)\\cdot\\:2\\sqrt{u}dv" }, { "type": "step", "primary": "$$v=\\sqrt{u}$$", "result": "=\\int\\:\\sec\\left(v\\right)\\tan\\left(v\\right)\\cdot\\:2vdv" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74MPJIqaqPxpkXK5pBiI3Cbyhjl8BhVkhQRW1MkI6xcWMuPq2OVC0VBJLHprrDOGMv+rpoTPOJyRosQJI2hFN+lsiU2T307EdVHE1M/3ZCa9LKrYeBoQQ0RAfHU9mKCZv/f/2HFzKslQpIZd7cfVvlvykvysQcMPKz0x3pB4VXMxTeQKHeh69S6dnv9vSoUoFJa4JWnJ2zIOyTZHz8ApdNP9dDfRMkV35qu7TRaGhBDX5B8JqzuKFmFVzV+OSdkVmQ==" } }, { "type": "step", "result": "=\\int\\:\\sec\\left(v\\right)\\tan\\left(v\\right)\\cdot\\:2vdv" }, { "type": "step", "primary": "Sacar la constante: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=2\\cdot\\:\\int\\:\\sec\\left(v\\right)\\tan\\left(v\\right)vdv" }, { "type": "interim", "title": "Aplicar integración por partes", "input": "\\int\\:\\sec\\left(v\\right)\\tan\\left(v\\right)vdv", "steps": [ { "type": "definition", "title": "Definición de integración por partes", "text": "$$\\int\\:uv'=uv-\\int\\:u'v$$" }, { "type": "step", "primary": "$$u=v$$" }, { "type": "step", "primary": "$$v'=\\sec\\left(v\\right)\\tan\\left(v\\right)$$" }, { "type": "interim", "title": "$$u'=\\frac{dv}{dv}=1$$", "input": "\\frac{dv}{dv}", "steps": [ { "type": "step", "primary": "Aplicar la regla de derivación: $$\\frac{dv}{dv}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgfyUWYtBrfkB3nVU/L275ZjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIaSUX15NRU/Cc5dZXPCJmBB017watY13TJdh++48Uzw8" } }, { "type": "interim", "title": "$$v=\\int\\:\\sec\\left(v\\right)\\tan\\left(v\\right)dv=\\sec\\left(v\\right)$$", "input": "\\int\\:\\sec\\left(v\\right)\\tan\\left(v\\right)dv", "steps": [ { "type": "interim", "title": "Aplicar integración por sustitución", "input": "\\int\\:\\sec\\left(v\\right)\\tan\\left(v\\right)dv", "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: $$w=\\sec\\left(v\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{dw}{dv}=\\sec\\left(v\\right)\\tan\\left(v\\right)$$", "input": "\\frac{d}{dv}\\left(\\sec\\left(v\\right)\\right)", "steps": [ { "type": "step", "primary": "Aplicar la regla de derivación: $$\\frac{d}{dv}\\left(\\sec\\left(v\\right)\\right)=\\sec\\left(v\\right)\\tan\\left(v\\right)$$", "result": "=\\sec\\left(v\\right)\\tan\\left(v\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkd4WE1uaK4l/McejooqQAz8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaug8chn5ff8mgKJWKE/HCvFOjeh7+jKEzLb7VNCEMF3Z/JBkKJWdpPanMBG+hWp1C1aFSxzDYtEOv7qB1/MfZ7A0=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dw=\\sec\\left(v\\right)\\tan\\left(v\\right)dv$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=\\frac{1}{\\sec\\left(v\\right)\\tan\\left(v\\right)}dw$$" }, { "type": "step", "result": "=\\int\\:w\\tan\\left(v\\right)\\frac{1}{\\sec\\left(v\\right)\\tan\\left(v\\right)}dw" }, { "type": "step", "primary": "$$w=\\sec\\left(v\\right)$$", "result": "=\\int\\:w\\tan\\left(v\\right)\\frac{1}{w\\tan\\left(v\\right)}dw" }, { "type": "interim", "title": "$$w\\tan\\left(v\\right)\\frac{1}{w\\tan\\left(v\\right)}=1$$", "input": "w\\tan\\left(v\\right)\\frac{1}{w\\tan\\left(v\\right)}", "steps": [ { "type": "step", "primary": "Multiplicar fracciones: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:w\\tan\\left(v\\right)}{w\\tan\\left(v\\right)}" }, { "type": "step", "primary": "Eliminar los terminos comunes: $$w$$", "result": "=\\frac{1\\cdot\\:\\tan\\left(v\\right)}{\\tan\\left(v\\right)}" }, { "type": "step", "primary": "Eliminar los terminos comunes: $$\\tan\\left(v\\right)$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mplVfI0JJQU5uws+wPv1C2Ti4wEYfBbqRFPv1+zC2CgtOtZYwUjyXhDTsNnn6ElrPGrllHeFjaKLCz4xpIb40ajjPiEWRGjZUqmBApxQcnJI14cFQql3tpLQIY6gqo1qHhL/3sg1S0NxsrxOoWu8kw==" } }, { "type": "step", "result": "=\\int\\:1dw" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s733VXqSvGRrlt9coPo6RRU8ZpSlgj74QlljnqOQpxvV7/6umhM84nJGixAkjaEU36WyJTZPfTsR1UcTUz/dkJr1JIOc5Ye/6BbhUGn8OeYwUZEt3ZXAiqUE0HIXrrrezJPsDL/c6fpjsyFQXRBaFTJqsBK8ljPcXhUzRU8e2AhQIJ4PsqTgzPzp9ryOMiXLLgCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\int\\:1dw" }, { "type": "step", "primary": "Integral de una constante: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:w" }, { "type": "step", "primary": "Sustituir en la ecuación $$w=\\sec\\left(v\\right)$$", "result": "=1\\cdot\\:\\sec\\left(v\\right)" }, { "type": "step", "primary": "Simplificar", "result": "=\\sec\\left(v\\right)", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Agregar una constante a la solución", "result": "=\\sec\\left(v\\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", "interimType": "Integrals" } }, { "type": "step", "result": "=v\\sec\\left(v\\right)-\\int\\:1\\cdot\\:\\sec\\left(v\\right)dv" }, { "type": "interim", "title": "Simplificar", "input": "v\\sec\\left(v\\right)-\\int\\:1\\cdot\\:\\sec\\left(v\\right)dv", "result": "=v\\sec\\left(v\\right)-\\int\\:\\sec\\left(v\\right)dv", "steps": [ { "type": "step", "primary": "Multiplicar: $$1\\cdot\\:\\sec\\left(v\\right)=\\sec\\left(v\\right)$$", "result": "=v\\sec\\left(v\\right)-\\int\\:\\sec\\left(v\\right)dv" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } } ], "meta": { "interimType": "Integration By Parts 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s733VXqSvGRrlt9coPo6RRU/eKwzcVZ+mBrbzbLbVKUhPsWjs8PnB44GqVUE5DCEHG78Zc2PnuzPhHJgIs2U9f2+Afbv2i/jpf8oyoC0yQxa/zWh97cbGTfLHve0ulJqsjhrSXXugAO6JR/Gk5CeOEK5N5Aod6Hr1Lp2e/29KhSgUlrglacnbMg7JNkfPwCl00wPLFnAMWuSCgwsuvs5k/DqwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=2\\left(v\\sec\\left(v\\right)-\\int\\:\\sec\\left(v\\right)dv\\right)" }, { "type": "interim", "title": "$$\\int\\:\\sec\\left(v\\right)dv=\\ln\\left|\\tan\\left(v\\right)+\\sec\\left(v\\right)\\right|$$", "input": "\\int\\:\\sec\\left(v\\right)dv", "steps": [ { "type": "step", "primary": "Aplicar la regla de integración: $$\\int\\:\\sec\\left(v\\right)dv=\\ln\\left|\\tan\\left(v\\right)+\\sec\\left(v\\right)\\right|$$", "result": "=\\ln\\left|\\tan\\left(v\\right)+\\sec\\left(v\\right)\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73x0cbOTxf0UQkB9YaET5hC4aB/JGgqfe8JH01ZhliKXpesMAscueMuiRPtRKMOipjKTYcRqKLK1nP3JD+iOGUCRrKJ/qNoT2ldOu5qKYTfQ/1U6sUwtyrqp1rYa50479Mp0H8D9a3Ioe8zuZotRYTsK0n+5nogBTmVp36CXtACxg86xSpXLbQsdDAD1d9X+P+4ADd7tUmJ6mD6Z3t28XUc=" } }, { "type": "step", "result": "=2\\left(v\\sec\\left(v\\right)-\\ln\\left|\\tan\\left(v\\right)+\\sec\\left(v\\right)\\right|\\right)" }, { "type": "interim", "title": "Sustitución hacia atrás", "input": "2\\left(v\\sec\\left(v\\right)-\\ln\\left|\\tan\\left(v\\right)+\\sec\\left(v\\right)\\right|\\right)", "result": "=2\\left(\\sqrt{x+1}\\sec\\left(\\sqrt{x+1}\\right)-\\ln\\left|\\tan\\left(\\sqrt{x+1}\\right)+\\sec\\left(\\sqrt{x+1}\\right)\\right|\\right)", "steps": [ { "type": "step", "primary": "Sustituir en la ecuación $$v=\\sqrt{u}$$", "result": "=2\\left(\\sqrt{u}\\sec\\left(\\sqrt{u}\\right)-\\ln\\left|\\tan\\left(\\sqrt{u}\\right)+\\sec\\left(\\sqrt{u}\\right)\\right|\\right)" }, { "type": "step", "primary": "Sustituir en la ecuación $$u=x+1$$", "result": "=2\\left(\\sqrt{x+1}\\sec\\left(\\sqrt{x+1}\\right)-\\ln\\left|\\tan\\left(\\sqrt{x+1}\\right)+\\sec\\left(\\sqrt{x+1}\\right)\\right|\\right)" } ], "meta": { "interimType": "Generic Substitute Back 0Eq" } }, { "type": "step", "primary": "Agregar una constante a la solución", "result": "=2\\left(\\sqrt{x+1}\\sec\\left(\\sqrt{x+1}\\right)-\\ln\\left|\\tan\\left(\\sqrt{x+1}\\right)+\\sec\\left(\\sqrt{x+1}\\right)\\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#area=main&subtopic=Integration%20By%20Parts", "practiceTopic": "Integration by Parts" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=2(\\sqrt{x+1}\\sec(\\sqrt{x+1})-\\ln\\left|\\tan(\\sqrt{x+1})+\\sec(\\sqrt{x+1})\\right|)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }