integral of 1/(y^6)-9/(sqrt(y))

{ "query": { "display": "$$\\int\\:\\frac{1}{y^{6}}-\\frac{9}{\\sqrt{y}}dy$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{y^{6}}-\\frac{9}{\\sqrt{y}}dy" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{1}{5y^{5}}-18\\sqrt{y}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1}{y^{6}}-\\frac{9}{\\sqrt{y}}dy=-\\frac{1}{5y^{5}}-18\\sqrt{y}+C$$", "input": "\\int\\:\\frac{1}{y^{6}}-\\frac{9}{\\sqrt{y}}dy", "steps": [ { "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{1}{y^{6}}dy-\\int\\:\\frac{9}{\\sqrt{y}}dy" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{y^{6}}dy=-\\frac{1}{5y^{5}}$$", "input": "\\int\\:\\frac{1}{y^{6}}dy", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\frac{1}{y^{6}}dy", "result": "=-\\frac{1}{5y^{5}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{y^{6}}=y^{-6}$$" ], "result": "=\\int\\:y^{-6}dy", "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{y^{-6+1}}{-6+1}" }, { "type": "interim", "title": "Simplify $$\\frac{y^{-6+1}}{-6+1}:{\\quad}-\\frac{1}{5y^{5}}$$", "input": "\\frac{y^{-6+1}}{-6+1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-6+1=-5$$", "result": "=\\frac{y^{-5}}{-5}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{y^{-5}}{5}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$y^{-5}=\\frac{1}{y^{5}}$$" ], "result": "=-\\frac{\\frac{1}{y^{5}}}{5}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{1}{y^{5}}}{5}=\\frac{1}{y^{5}\\cdot\\:5}$$" ], "result": "=-\\frac{1}{y^{5}\\cdot\\:5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{1}{5y^{5}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s72WfHkDE1YACKSrrekktqfEqzJGeyM6S0DCrZisZSya8rrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx4OQzbEnsuafNY3nX9QxDlJSIo/P8EGCOXQKR+9GBVsIXql8XXPq6bNQlMm+36iNhkM590FLV/JQkTs6OcJgAWaNH+Jl/Zyy+v2DeiqbV+Z5w==" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{9}{\\sqrt{y}}dy=18\\sqrt{y}$$", "input": "\\int\\:\\frac{9}{\\sqrt{y}}dy", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=9\\cdot\\:\\int\\:\\frac{1}{\\sqrt{y}}dy" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\frac{1}{\\sqrt{y}}dy", "result": "=9\\cdot\\:2y^{\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\int\\:\\frac{1}{y^{\\frac{1}{2}}}dy", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{y^{\\frac{1}{2}}}=y^{-\\frac{1}{2}}$$" ], "result": "=\\int\\:y^{-\\frac{1}{2}}dy", "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{y^{-\\frac{1}{2}+1}}{-\\frac{1}{2}+1}" }, { "type": "interim", "title": "Simplify $$\\frac{y^{-\\frac{1}{2}+1}}{-\\frac{1}{2}+1}:{\\quad}2y^{\\frac{1}{2}}$$", "input": "\\frac{y^{-\\frac{1}{2}+1}}{-\\frac{1}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$-\\frac{1}{2}+1:{\\quad}\\frac{1}{2}$$", "input": "-\\frac{1}{2}+1", "result": "=\\frac{y^{-\\frac{1}{2}+1}}{\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1}{2}+\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$-1+1\\cdot\\:2=1$$", "input": "-1+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-1+2" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+2=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vn6CQNwxYfV8gDG5KuxmLlXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qk4XEODzDfV3RCKyB8F0QybUZ0qCI2z4+iUq9O5bNiT8=" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "$$y^{-\\frac{1}{2}+1}=y^{\\frac{1}{2}}$$", "input": "y^{-\\frac{1}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$-\\frac{1}{2}+1:{\\quad}\\frac{1}{2}$$", "input": "-\\frac{1}{2}+1", "result": "=y^{\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1}{2}+\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$-1+1\\cdot\\:2=1$$", "input": "-1+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-1+2" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+2=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vn6CQNwxYfV8gDG5KuxmLlXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qk4XEODzDfV3RCKyB8F0QybUZ0qCI2z4+iUq9O5bNiT8=" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lFtozWTrJ4ZHShE8HU01xqL3R/dXk768t6JSjGXJZIJwkKGJWEPFPk38sdJMsyPImBQrc8o7HuGJD0ypiKqeAwH2kDe5DGYTz3TrPquGdIh2pktA4t50lLXx8qSqGcZzMpjNV8PpUg5e+keZOfJaclQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{y^{\\frac{1}{2}}}{\\frac{1}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{y^{\\frac{1}{2}}\\cdot\\:2}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=2y^{\\frac{1}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=2y^{\\frac{1}{2}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73YBJdU3jgZIBfx9KjN2Y10fcFuztWIIoL1T3musx6jiOiaLJuL5RxgumX0gNvT19evIf9R89K46QJJ+28FMIXjMnQ++uQyqJzUIcr7WpYOZbdO+34F7CgaMka2kBYb/doEFMST8lDZxn1Yq5HMKVTsN/SZgQjH1OoahVjOHG2Hrj8LNU2fafRgGTDrnDOEnog==" } }, { "type": "interim", "title": "Simplify $$9\\cdot\\:2y^{\\frac{1}{2}}:{\\quad}18\\sqrt{y}$$", "input": "9\\cdot\\:2y^{\\frac{1}{2}}", "result": "=18\\sqrt{y}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$9\\cdot\\:2=18$$", "result": "=18y^{\\frac{1}{2}}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "result": "=18\\sqrt{y}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7d15UOXCp1pNB0Z9I1Dnk8ZQUBbYf03uuUWubkbKEUk7dd47a0hQ8flDbGsI5To1dj0xb3LEj34QCVqNOaJiC4DamwaZ/9iLvGF2RbpiN6RHuQCM/vqpbrqU5SxRHBPSd2SKIAGCU9hd3xviSqaQd2C4fJ1D0nCRvpCEVBdWP6M6/Mg94S0N9we//Py6WzxN6" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-\\frac{1}{5y^{5}}-18\\sqrt{y}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{1}{5y^{5}}-18\\sqrt{y}+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=Sum%20Rule", "practiceTopic": "Integral Sum Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "y", "plotRequest": "y=-\\frac{1}{5y^{5}}-18\\sqrt{y}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }