integral from 0 to infinity of 1/(e^x-1)

{ "query": { "display": "$$\\int_{0}^{\\infty\\:}\\frac{1}{e^{x}-1}dx$$", "symbolab_question": "BIG_OPERATOR#\\int _{0}^{\\infty }\\frac{1}{e^{x}-1}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Definite Integrals", "default": "\\mathrm{diverges}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int_{0}^{\\infty\\:}\\frac{1}{e^{x}-1}dx=$$diverges", "input": "\\int_{0}^{\\infty\\:}\\frac{1}{e^{x}-1}dx", "steps": [ { "type": "interim", "title": "Compute the indefinite integral:$${\\quad}\\int\\:\\frac{1}{e^{x}-1}dx=\\ln\\left|e^{x}-1\\right|-x+C$$", "input": "\\int\\:\\frac{1}{e^{x}-1}dx", "steps": [ { "type": "interim", "title": "$$\\frac{1}{e^{x}-1}=\\frac{e^{x}}{e^{x}-1}-1$$", "input": "\\frac{1}{e^{x}-1}", "steps": [ { "type": "step", "primary": "$$\\frac{1}{e^{x}-1}=\\frac{1+\\left(e^{x}-1\\right)}{e^{x}-1}-1$$", "result": "=\\frac{1+\\left(e^{x}-1\\right)}{e^{x}-1}-1", "meta": { "title": { "extension": "Apply the following algebraic property$$:{\\quad}\\frac{a}{1-a}=\\frac{1}{1-a}-1$$<br/>$$\\frac{a}{1-a}=\\frac{1-1+a}{1-a}=\\frac{1}{1-a}+\\frac{-1+a}{1-a}=\\frac{1}{1-a}+\\frac{-\\left(1-a\\right)}{1-a}=\\frac{1}{1-a}-1$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{1+\\left(e^{x}-1\\right)}{e^{x}-1}-1:{\\quad}\\frac{e^{x}}{e^{x}-1}-1$$", "input": "\\frac{1+\\left(e^{x}-1\\right)}{e^{x}-1}-1", "result": "=\\frac{e^{x}}{e^{x}-1}-1", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=\\frac{1+e^{x}-1}{e^{x}-1}-1" }, { "type": "interim", "title": "$$1+e^{x}-1=e^{x}$$", "input": "1+e^{x}-1", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=e^{x}+1-1" }, { "type": "step", "primary": "$$1-1=0$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73CTQEgq4U/wwPPt3my/tcN13jtrSFDx+UNsawjlOjV1HOCqrfTgpePcNI5r/X52yIT9snT30d0/IETJbq0x/vMxYVkKYTUQkOfqbOY7Y2E8=" } }, { "type": "step", "result": "=\\frac{e^{x}}{e^{x}-1}-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\int\\:\\frac{e^{x}}{e^{x}-1}-1dx" }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=\\int\\:\\frac{e^{x}}{e^{x}-1}dx-\\int\\:1dx" }, { "type": "interim", "title": "$$\\int\\:\\frac{e^{x}}{e^{x}-1}dx=\\ln\\left|e^{x}-1\\right|$$", "input": "\\int\\:\\frac{e^{x}}{e^{x}-1}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{e^{x}}{e^{x}-1}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "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": [ "Substitute: $$u=e^{x}-1$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=e^{x}$$", "input": "\\frac{d}{dx}\\left(e^{x}-1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(e^{x}\\right)-\\frac{d}{dx}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "input": "\\frac{d}{dx}\\left(e^{x}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(e^{x}\\right)=e^{x}$$", "result": "=e^{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjqxKAa6SkUhZrTPjmns35ik3hxk9aCfAWodBRxXgUexthpiW0WhiZGad41dobHknD/L0MoYg+CUn6oyL3EO7YobH0DM3/kWbWNIKRYne2WotxxYM+bRuETn+hrcvGmLsg==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "step", "result": "=e^{x}-0" }, { "type": "step", "primary": "Simplify", "result": "=e^{x}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=e^{x}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{e^{x}}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{e^{x}}{u}\\cdot\\:\\frac{1}{e^{x}}du" }, { "type": "interim", "title": "Simplify $$\\frac{e^{x}}{u}\\cdot\\:\\frac{1}{e^{x}}:{\\quad}\\frac{1}{u}$$", "input": "\\frac{e^{x}}{u}\\cdot\\:\\frac{1}{e^{x}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{e^{x}\\cdot\\:1}{ue^{x}}" }, { "type": "step", "primary": "Cancel the common factor: $$e^{x}$$", "result": "=\\frac{1}{u}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7wASI1sIbyM7JUZ22HIzvtvFPtUs/HkMvYCes+Ia5pCS+qPBBXW0OBa8HfjFPmx5EoAIezcf2HGXjCQp0SyS6as7d3vw7eocUrI0TgQGIqC/JQmO8m76yr+ykT+C81F97XZ04JX0uaeiV4M8qp8wkzZ9DlD8l0LRRojo5hK/1zTOxqM8sxPr6KIJhDsVwKxEeyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u}du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u}du=\\ln\\left(\\left|u\\right|\\right)$$", "result": "=\\ln\\left|u\\right|" }, { "type": "step", "primary": "Substitute back $$u=e^{x}-1$$", "result": "=\\ln\\left|e^{x}-1\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:1dx=x$$", "input": "\\int\\:1dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\ln\\left|e^{x}-1\\right|-x" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\ln\\left|e^{x}-1\\right|-x+C", "meta": { "title": { "extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$" } } } ], "meta": { "solvingClass": "Integrals", "interimType": "Definite Integral For Indefinite Title 0Eq" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\int_{0}^{\\infty\\:}\\frac{1}{e^{x}-1}dx=$$diverges", "result": "=\\mathrm{diverges}", "steps": [ { "type": "step", "primary": "$$\\int_{a}^{b}{f\\left(x\\right)dx}=F\\left(b\\right)-F\\left(a\\right)=\\lim_{x\\to\\:b-}\\left(F\\left(x\\right)\\right)-\\lim_{x\\to\\:a+}\\left(F\\left(x\\right)\\right)$$" }, { "type": "step", "primary": "Since $$\\lim_{x\\to\\:\\infty\\:}\\left(\\ln\\left|e^{x}-1\\right|-x\\right)=\\infty\\:$$", "result": "=\\mathrm{diverges}" } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq" } } ] }, "meta": { "showVerify": true } }