integral of e^{-x}(9+e^{-x})

{ "query": { "display": "$$\\int\\:e^{-x}\\left(9+e^{-x}\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int e^{-x}(9+e^{-x})dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{(9+e^{-x})^{2}}{2}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:e^{-x}\\left(9+e^{-x}\\right)dx=-\\frac{\\left(9+e^{-x}\\right)^{2}}{2}+C$$", "input": "\\int\\:e^{-x}\\left(9+e^{-x}\\right)dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:e^{-x}\\left(9+e^{-x}\\right)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: $$u=9+e^{-x}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=-e^{-x}$$", "input": "\\frac{d}{dx}\\left(9+e^{-x}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(9\\right)+\\frac{d}{dx}\\left(e^{-x}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(9\\right)=0$$", "input": "\\frac{d}{dx}\\left(9\\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+idP5vLVrjUll6eMdYvTYt8V2vaKd9AkmdqaaXItJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsdQFf8ULyNiNWh+aczQhPx" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(e^{-x}\\right)=-e^{-x}$$", "input": "\\frac{d}{dx}\\left(e^{-x}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}e^{-x}\\frac{d}{dx}\\left(-x\\right)$$", "input": "\\frac{d}{dx}\\left(e^{-x}\\right)", "result": "=e^{-x}\\frac{d}{dx}\\left(-x\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=e^{u},\\:\\:u=-x$$" ], "result": "=\\frac{d}{du}\\left(e^{u}\\right)\\frac{d}{dx}\\left(-x\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "input": "\\frac{d}{du}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqCr3EWRZw3L4+rHTTdVG0Ok3hxk9aCfAWodBRxXgUexwx+RE9MtjN5hKMwTI7fffj/L0MoYg+CUn6oyL3EO7YrHahlpzKGY893KZ4T4i4Tv3RCXWsqiNx7T9zOhL5sYfw==" } }, { "type": "step", "result": "=e^{u}\\frac{d}{dx}\\left(-x\\right)" }, { "type": "step", "primary": "Substitute back $$u=-x$$", "result": "=e^{-x}\\frac{d}{dx}\\left(-x\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmBepZeKV8bVrdHbYZEMXtOTdaV09PMxEKZ9FieghTFwc6p4sNpoW3XPzo2W3Ux6aosjLe8tD9HbrkG8vq6q9jiWR6DAIfSwNblE3ziU9D9b9KdAd3NQb43TJHJRafLwY/C30sSftAIFS6Qkpy19IkrNs0uRhmwmWtV92tk+c/8Zu/mDTHcAziAiYeNOzjloJg==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(-x\\right)=-1$$", "input": "\\frac{d}{dx}\\left(-x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=-\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYlIFiSG67zLdq9NRnof0E3fZGku9zFkxwe1dTH8vycb9+906N5fPsfVzu/TlcIEC4c1bIZxfodm3UsZcfZAZr4tZsYzC8RJ0SqIyNjewPF2dJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=e^{-x}\\left(-1\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-e^{-x}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=0-e^{-x}" }, { "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": "$$\\quad\\Rightarrow\\:dx=\\left(-e^{x}\\right)du$$" }, { "type": "step", "result": "=\\int\\:e^{-x}u\\left(-e^{x}\\right)du" }, { "type": "interim", "title": "Simplify $$e^{-x}u\\left(-e^{x}\\right):{\\quad}-u$$", "input": "e^{-x}u\\left(-e^{x}\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-e^{-x}ue^{x}" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$e^{-x}e^{x}=\\:e^{-x+x}$$" ], "result": "=-ue^{-x+x}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add similar elements: $$-x+x=0$$", "result": "=-ue^{0}" }, { "type": "step", "primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$", "result": "=-1\\cdot\\:u" }, { "type": "step", "primary": "Multiply: $$u\\cdot\\:1=u$$", "result": "=-u" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:-udu" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s76lg3ebd/YWy6iKCI/g2wi+r+0zX0wqSmAVtc7NV8L0A/6umhM84nJGixAkjaEU36WyJTZPfTsR1UcTUz/dkJr0tLoVIh4O4ahvnBYDckgF7dnTglfS5p6JXgzyqnzCTNn0OUPyXQtFGiOjmEr/XNM7GozyzE+voogmEOxXArER7JLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=\\int\\:-udu" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-\\int\\:udu" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:udu", "result": "=-\\frac{u^{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{u^{1+1}}{1+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{1+1}}{1+1}:{\\quad}\\frac{u^{2}}{2}$$", "input": "\\frac{u^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{u^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{u^{2}}{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77irOeniMfrJKKN+TrhAcvL/JyKXuO90NgYuEtRnVFUoQEgTxsQDcbkC7lns/WqbpAUgTzPSdH5PWV4NCtvwjA7/YrZ1UCh4L70vx5eDNyDLTeQKHeh69S6dnv9vSoUoFEMybLZHp2MhZ1cw+jOu7RuDCZKz/+DESbePVmsYY2Aq" } }, { "type": "step", "primary": "Substitute back $$u=9+e^{-x}$$", "result": "=-\\frac{\\left(9+e^{-x}\\right)^{2}}{2}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{\\left(9+e^{-x}\\right)^{2}}{2}+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": "x", "plotRequest": "y=-\\frac{(9+e^{-x})^{2}}{2}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }