integral of x^{1/7}

{ "query": { "display": "$$\\int\\:x^{\\frac{1}{7}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int x^{\\frac{1}{7}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{7}{8}x^{\\frac{8}{7}}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:x^{\\frac{1}{7}}dx=\\frac{7}{8}x^{\\frac{8}{7}}+C$$", "input": "\\int\\:x^{\\frac{1}{7}}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{\\frac{1}{7}}dx", "result": "=\\frac{7}{8}x^{\\frac{8}{7}}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{\\frac{1}{7}+1}}{\\frac{1}{7}+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{\\frac{1}{7}+1}}{\\frac{1}{7}+1}:{\\quad}\\frac{7}{8}x^{\\frac{8}{7}}$$", "input": "\\frac{x^{\\frac{1}{7}+1}}{\\frac{1}{7}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{7}+1:{\\quad}\\frac{8}{7}$$", "input": "\\frac{1}{7}+1", "result": "=\\frac{x^{\\frac{1}{7}+1}}{\\frac{8}{7}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:7}{7}$$", "result": "=\\frac{1}{7}+\\frac{1\\cdot\\:7}{7}" }, { "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\\:7}{7}" }, { "type": "interim", "title": "$$1+1\\cdot\\:7=8$$", "input": "1+1\\cdot\\:7", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:7=7$$", "result": "=1+7" }, { "type": "step", "primary": "Add the numbers: $$1+7=8$$", "result": "=8" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7spTvd1WiV52ZJymrXr8Zfd6GQqufR6tr2vPxOUv7H++QuJfcIt8gbvzPBjbmbYE2Xx2gq2/8uoBg1ahOTmc2TO9m/kQb41HiwJAexTs6Jn0=" } }, { "type": "step", "result": "=\\frac{8}{7}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "$$x^{\\frac{1}{7}+1}=x^{\\frac{8}{7}}$$", "input": "x^{\\frac{1}{7}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{7}+1:{\\quad}\\frac{8}{7}$$", "input": "\\frac{1}{7}+1", "result": "=x^{\\frac{8}{7}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:7}{7}$$", "result": "=\\frac{1}{7}+\\frac{1\\cdot\\:7}{7}" }, { "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\\:7}{7}" }, { "type": "interim", "title": "$$1+1\\cdot\\:7=8$$", "input": "1+1\\cdot\\:7", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:7=7$$", "result": "=1+7" }, { "type": "step", "primary": "Add the numbers: $$1+7=8$$", "result": "=8" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7spTvd1WiV52ZJymrXr8Zfd6GQqufR6tr2vPxOUv7H++QuJfcIt8gbvzPBjbmbYE2Xx2gq2/8uoBg1ahOTmc2TO9m/kQb41HiwJAexTs6Jn0=" } }, { "type": "step", "result": "=\\frac{8}{7}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AmASmqccjHpJZQtWBzjYsSa+StGnWtengtTZdLN8SGHMwViaLUXkeD+JukROhWdjiTZ26kB51WaL3u2Fw+n4Cv8//6/nV5O4fb8Xgwi7maq2n8dNrwOWuvOTedK5iPzdOSmYyF7/9WD0pgkI6N37O1SToOgbHFGM6g87Y3Z7KlM=" } }, { "type": "step", "result": "=\\frac{x^{\\frac{8}{7}}}{\\frac{8}{7}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{x^{\\frac{8}{7}}\\cdot\\:7}{8}" }, { "type": "step", "result": "=\\frac{7}{8}x^{\\frac{8}{7}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{7}{8}x^{\\frac{8}{7}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s76pEGTFZ6OPGVSy3TVMV5+Wr+0zX0wqSmAVtc7NV8L0Arrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx6Gd8O4OGnJhdqAuPbTVOpUv/1AMpiengo7x/DpZnEgsR9tkjxmSv8J4QuSfftN+UVSBv6izheLVUKQ/emokAUyVi4djWqB/aQf/oQfiXX55rCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{7}{8}x^{\\frac{8}{7}}+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=Power%20Rule", "practiceTopic": "Integral Power Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{7}{8}x^{\\frac{8}{7}}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }