integral of e^y(e^y+1)^{-2}

{ "query": { "display": "$$\\int\\:e^{y}\\left(e^{y}+1\\right)^{-2}dy$$", "symbolab_question": "BIG_OPERATOR#\\int e^{y}(e^{y}+1)^{-2}dy" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{1}{e^{y}+1}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:e^{y}\\left(e^{y}+1\\right)^{-2}dy=-\\frac{1}{e^{y}+1}+C$$", "input": "\\int\\:e^{y}\\left(e^{y}+1\\right)^{-2}dy", "steps": [ { "type": "interim", "title": "Simplify $$e^{y}\\left(e^{y}+1\\right)^{-2}:{\\quad}\\frac{e^{y}}{\\left(e^{y}+1\\right)^{2}}$$", "input": "e^{y}\\left(e^{y}+1\\right)^{-2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$\\left(e^{y}+1\\right)^{-2}=\\frac{1}{\\left(e^{y}+1\\right)^{2}}$$" ], "result": "=e^{y}\\frac{1}{\\left(e^{y}+1\\right)^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:e^{y}}{\\left(e^{y}+1\\right)^{2}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:e^{y}=e^{y}$$", "result": "=\\frac{e^{y}}{\\left(e^{y}+1\\right)^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{e^{y}}{\\left(e^{y}+1\\right)^{2}}dy" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{e^{y}}{\\left(e^{y}+1\\right)^{2}}dy", "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: $$u=e^{y}+1$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dy}=e^{y}$$", "input": "\\frac{d}{dy}\\left(e^{y}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dy}\\left(e^{y}\\right)+\\frac{d}{dy}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dy}\\left(e^{y}\\right)=e^{y}$$", "input": "\\frac{d}{dy}\\left(e^{y}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dy}\\left(e^{y}\\right)=e^{y}$$", "result": "=e^{y}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuXAxmneY4KHTzKqTj/QvQik3hxk9aCfAWodBRxXgUexn8DvPuo3s1Tjjq0ztCM2mj/L0MoYg+CUn6oyL3EO7Yrh1NTqo1IauVC4dJ+9+pNcVrQLAn3CCpdhAcVfmnOKQw==" } }, { "type": "interim", "title": "$$\\frac{d}{dy}\\left(1\\right)=0$$", "input": "\\frac{d}{dy}\\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+idP5vLVrjUll6eMdYjo/Vfk8lzzNRp1PIi2YEfNJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTvtPdL9kNVzLtZYJUOd2Nd5" } }, { "type": "step", "result": "=e^{y}+0" }, { "type": "step", "primary": "Simplify", "result": "=e^{y}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=e^{y}dy$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dy=\\frac{1}{e^{y}}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{e^{y}}{u^{2}}\\cdot\\:\\frac{1}{e^{y}}du" }, { "type": "interim", "title": "Simplify $$\\frac{e^{y}}{u^{2}}\\cdot\\:\\frac{1}{e^{y}}:{\\quad}\\frac{1}{u^{2}}$$", "input": "\\frac{e^{y}}{u^{2}}\\cdot\\:\\frac{1}{e^{y}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{e^{y}\\cdot\\:1}{u^{2}e^{y}}" }, { "type": "step", "primary": "Cancel the common factor: $$e^{y}$$", "result": "=\\frac{1}{u^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u^{2}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s72nDKVkElyTlIIFdkNhULG3cRUGfyvL3vVQNE3zBT75lpN4cZPWgnwFqHQUcV4FHsfBLh5j/jJcd1Frv9s/1xSw0pWMfsJc1e/Z0+a/wFZqiVdawfdGtW4aUXHrK9DlWLU/jhwLaEyoMcSFP2OCr+tbwt9LEn7QCBUukJKctfSJKZQn2SiIEHPukOBxMduK6is14g2py5PWS6zSGkqNBJTQ=" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{u^{2}}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\frac{1}{u^{2}}du", "result": "=-\\frac{1}{u}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{u^{2}}=u^{-2}$$" ], "result": "=\\int\\:u^{-2}du", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{u^{-2+1}}{-2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{-2+1}}{-2+1}:{\\quad}-\\frac{1}{u}$$", "input": "\\frac{u^{-2+1}}{-2+1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=\\frac{u^{-1}}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{u^{-1}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-u^{-1}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-1}=\\frac{1}{a}$$", "result": "=-\\frac{1}{u}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{1}{u}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/RSr02Agv0MR/qV7Nm+eMMy4+rY5ULRUEksemusM4Yyrrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx4OQzbEnsuafNY3nX9QxDlJ1HXTSqqQEjS1gpf6I+JyHQS4M5VpC8qh+oehjmM1qmweKkh+28FiXwy+Vsz8xLQiialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "Substitute back $$u=e^{y}+1$$", "result": "=-\\frac{1}{e^{y}+1}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{1}{e^{y}+1}+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", "practiceLink": "/practice/integration-practice#area=main&subtopic=Substitution", "practiceTopic": "Integral Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "y", "plotRequest": "y=-\\frac{1}{e^{y}+1}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }