integral of (x^2-2)sqrt(x)

{ "query": { "display": "$$\\int\\:\\left(x^{2}-2\\right)\\sqrt{x}dx$$", "symbolab_question": "BIG_OPERATOR#\\int (x^{2}-2)\\sqrt{x}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{2}{7}x^{\\frac{7}{2}}-\\frac{4}{3}x^{\\frac{3}{2}}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\left(x^{2}-2\\right)\\sqrt{x}dx=\\frac{2}{7}x^{\\frac{7}{2}}-\\frac{4}{3}x^{\\frac{3}{2}}+C$$", "input": "\\int\\:\\left(x^{2}-2\\right)\\sqrt{x}dx", "steps": [ { "type": "interim", "title": "Expand $$\\left(x^{2}-2\\right)\\sqrt{x}:{\\quad}x^{\\frac{5}{2}}-2\\sqrt{x}$$", "input": "\\left(x^{2}-2\\right)\\sqrt{x}", "steps": [ { "type": "step", "result": "=\\sqrt{x}\\left(x^{2}-2\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=\\sqrt{x},\\:b=x^{2},\\:c=2$$" ], "result": "=\\sqrt{x}x^{2}-\\sqrt{x}\\cdot\\:2", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=x^{2}\\sqrt{x}-2\\sqrt{x}" }, { "type": "interim", "title": "$$x^{2}\\sqrt{x}=x^{\\frac{5}{2}}$$", "input": "x^{2}\\sqrt{x}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}\\sqrt{x}=\\:x^{2}x^{\\frac{1}{2}}=\\:x^{2+\\frac{1}{2}}$$" ], "result": "=x^{2+\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Join $$2+\\frac{1}{2}:{\\quad}\\frac{5}{2}$$", "input": "2+\\frac{1}{2}", "result": "=x^{\\frac{5}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$2=\\frac{2\\cdot\\:2}{2}$$", "result": "=\\frac{2\\cdot\\:2}{2}+\\frac{1}{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{2\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$2\\cdot\\:2+1=5$$", "input": "2\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4+1" }, { "type": "step", "primary": "Add the numbers: $$4+1=5$$", "result": "=5" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7enR8wmwCBefJD9FTK9S3Ut6GQqufR6tr2vPxOUv7H+9Ao0ShslLSdH/VCTigfQH+1SvY/eJGzvEmlW7hoPETDk8GCUzRs5gGR0iiXlZ950Y=" } }, { "type": "step", "result": "=\\frac{5}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7uOAIiXlP8+vEB7cCsZWnTACWKUbvV6WK3fDUgFtg3Q95OLASxP5X2qjwGWBXK/F88nwqfP3qaPIixX4c2FG0vf8obwNncpqNs2rFLHrcSDf4cerPxBPTEldRofTioSTU0F6NlOqiS4Ige/xxPpts1Q==" } }, { "type": "step", "result": "=x^{\\frac{5}{2}}-2\\sqrt{x}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hMO17e2Yit0qDO9ZS6topgDwo3FKGG+l60YfOubcQ5fMwViaLUXkeD+JukROhWdjqpO7VabXUp19fZIbVBt/1FDfpBNQIfjWw7Cy0RDD22nvbBmbuQNTF0TphKZ8RuvaPBE1ZmKXh9mB0Zbk2zX9xUJxVy0wVftBc/klLiE7lPa/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "=\\int\\:x^{\\frac{5}{2}}-2\\sqrt{x}dx" }, { "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\\:x^{\\frac{5}{2}}dx-\\int\\:2\\sqrt{x}dx" }, { "type": "interim", "title": "$$\\int\\:x^{\\frac{5}{2}}dx=\\frac{2}{7}x^{\\frac{7}{2}}$$", "input": "\\int\\:x^{\\frac{5}{2}}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{\\frac{5}{2}}dx", "result": "=\\frac{2}{7}x^{\\frac{7}{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{x^{\\frac{5}{2}+1}}{\\frac{5}{2}+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{\\frac{5}{2}+1}}{\\frac{5}{2}+1}:{\\quad}\\frac{2}{7}x^{\\frac{7}{2}}$$", "input": "\\frac{x^{\\frac{5}{2}+1}}{\\frac{5}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{5}{2}+1:{\\quad}\\frac{7}{2}$$", "input": "\\frac{5}{2}+1", "result": "=\\frac{x^{\\frac{5}{2}+1}}{\\frac{7}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{5}{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{5+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$5+1\\cdot\\:2=7$$", "input": "5+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=5+2" }, { "type": "step", "primary": "Add the numbers: $$5+2=7$$", "result": "=7" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NkTDa0KoYbiuF02zzEqEbd6GQqufR6tr2vPxOUv7H+9MTg4418YnsnbKpNwPhLdufctDmfCrBrJLusHV8SYEn4swtBHn3Smw9y+93IXxzSc=" } }, { "type": "step", "result": "=\\frac{7}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "$$x^{\\frac{5}{2}+1}=x^{\\frac{7}{2}}$$", "input": "x^{\\frac{5}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{5}{2}+1:{\\quad}\\frac{7}{2}$$", "input": "\\frac{5}{2}+1", "result": "=x^{\\frac{7}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{5}{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{5+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$5+1\\cdot\\:2=7$$", "input": "5+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=5+2" }, { "type": "step", "primary": "Add the numbers: $$5+2=7$$", "result": "=7" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NkTDa0KoYbiuF02zzEqEbd6GQqufR6tr2vPxOUv7H+9MTg4418YnsnbKpNwPhLdufctDmfCrBrJLusHV8SYEn4swtBHn3Smw9y+93IXxzSc=" } }, { "type": "step", "result": "=\\frac{7}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oRJK27Oh+uzOiT/TQq7/ySa+StGnWtengtTZdLN8SGHMwViaLUXkeD+JukROhWdjPX71FLgClH8jm/lHzsggn/8//6/nV5O4fb8Xgwi7mar2QdX0deCst2cuq72UJ7PJZUSdz5P/soKTm6NO4qC2uViVI3uvN1by+AN9NfjoKFU=" } }, { "type": "step", "result": "=\\frac{x^{\\frac{7}{2}}}{\\frac{7}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{x^{\\frac{7}{2}}\\cdot\\:2}{7}" }, { "type": "step", "result": "=\\frac{2}{7}x^{\\frac{7}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{2}{7}x^{\\frac{7}{2}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s71za2WXdiwjM306Px3iXza+r+0zX0wqSmAVtc7NV8L0Arrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx6JB7Asf8mi7IuDtv3AUYazfrOU3ZEv/Mj8dI6T6mWG+/mUz1GYa4snANpQYMxmVABSBv6izheLVUKQ/emokAUyVi4djWqB/aQf/oQfiXX55rCI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:2\\sqrt{x}dx=\\frac{4}{3}x^{\\frac{3}{2}}$$", "input": "\\int\\:2\\sqrt{x}dx", "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": "=2\\cdot\\:\\int\\:\\sqrt{x}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\sqrt{x}dx", "result": "=2\\cdot\\:\\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": "interim", "title": "Simplify $$2\\cdot\\:\\frac{2}{3}x^{\\frac{3}{2}}:{\\quad}\\frac{4}{3}x^{\\frac{3}{2}}$$", "input": "2\\cdot\\:\\frac{2}{3}x^{\\frac{3}{2}}", "result": "=\\frac{4}{3}x^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2\\cdot\\:2}{3}x^{\\frac{3}{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{4}{3}x^{\\frac{3}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviv4UQ5b6oL6EEyNvjoFFVldMhCvz+gotECPvNL5Xf6HZq47vuWedXv2WUg94ER8IwXZ+7cf1+YsRTaPa7cnp7ukDE/A7AGrvqmLaD33y/7sNZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2UQNuZdvAUkLju+GWp/hef/hRDlvqgvoQTI2+OgUVWV/4Os1kraVLsexDCmlrMF88=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{2}{7}x^{\\frac{7}{2}}-\\frac{4}{3}x^{\\frac{3}{2}}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{2}{7}x^{\\frac{7}{2}}-\\frac{4}{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#area=main&subtopic=Sum%20Rule", "practiceTopic": "Integral Sum Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{2}{7}x^{\\frac{7}{2}}-\\frac{4}{3}x^{\\frac{3}{2}}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }