integral of sqrt(x)

{ "query": { "display": "integral $$\\sqrt{x}$$", "symbolab_question": "PRE_CALC#integral \\sqrt{x}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{2}{3}x^{\\frac{3}{2}}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\sqrt{x}dx=\\frac{2}{3}x^{\\frac{3}{2}}+C$$", "input": "\\int\\:\\sqrt{x}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\sqrt{x}dx", "result": "=\\frac{2}{3}x^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\int\\:x^{\\frac{1}{2}}dx", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "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}{2}+1}}{\\frac{1}{2}+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}:{\\quad}\\frac{2}{3}x^{\\frac{3}{2}}$$", "input": "\\frac{x^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}+1:{\\quad}\\frac{3}{2}$$", "input": "\\frac{1}{2}+1", "result": "=\\frac{x^{\\frac{1}{2}+1}}{\\frac{3}{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": "$$1+1\\cdot\\:2=3$$", "input": "1+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=1+2" }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JDYTlBE4VraUkLRwnMmR2t6GQqufR6tr2vPxOUv7H++tj2bbSfUlSjnYqC784D/mXx2gq2/8uoBg1ahOTmc2TNo/74bofy09c99TpWFhG0I=" } }, { "type": "step", "result": "=\\frac{3}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "$$x^{\\frac{1}{2}+1}=x^{\\frac{3}{2}}$$", "input": "x^{\\frac{1}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}+1:{\\quad}\\frac{3}{2}$$", "input": "\\frac{1}{2}+1", "result": "=x^{\\frac{3}{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": "$$1+1\\cdot\\:2=3$$", "input": "1+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=1+2" }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JDYTlBE4VraUkLRwnMmR2t6GQqufR6tr2vPxOUv7H++tj2bbSfUlSjnYqC784D/mXx2gq2/8uoBg1ahOTmc2TNo/74bofy09c99TpWFhG0I=" } }, { "type": "step", "result": "=\\frac{3}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qijEBDcyPMwV4Y1jeiGyoCa+StGnWtengtTZdLN8SGHMwViaLUXkeD+JukROhWdj54FOx6X2WhOdSzB/njnZX/8//6/nV5O4fb8Xgwi7maommAxfeVI7cE/MHk5RAGQjmPksuCweRseg2ne4TzFM/1iVI3uvN1by+AN9NfjoKFU=" } }, { "type": "step", "result": "=\\frac{x^{\\frac{3}{2}}}{\\frac{3}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{x^{\\frac{3}{2}}\\cdot\\:2}{3}" }, { "type": "step", "result": "=\\frac{2}{3}x^{\\frac{3}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{2}{3}x^{\\frac{3}{2}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74k0AvuV0dGA6GA9LJJfPZ4sjvX7KVUO/AeCFSId4S33HipIftvBYl8MvlbM/MS0IniX35dQ/h01lIvxamZtt5PJRlyoDNDpRXyTjBcaa12uxESrjjaZRaDGtyZzqQyBzoEFMST8lDZxn1Yq5HMKVTsN/SZgQjH1OoahVjOHG2Hrj8LNU2fafRgGTDrnDOEnog==" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{2}{3}x^{\\frac{3}{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", "practiceTopic": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{2}{3}x^{\\frac{3}{2}}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }