integral of (sqrt(y^2-49))/y

{ "query": { "display": "$$\\int\\:\\frac{\\sqrt{y^{2}-49}}{y}dy$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{\\sqrt{y^{2}-49}}{y}dy" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-7\\arcsec(\\frac{1}{7}y)+\\sqrt{y^{2}-49}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{\\sqrt{y^{2}-49}}{y}dy=-7\\arcsec\\left(\\frac{1}{7}y\\right)+\\sqrt{y^{2}-49}+C$$", "input": "\\int\\:\\frac{\\sqrt{y^{2}-49}}{y}dy", "steps": [ { "type": "interim", "title": "Apply Trigonometric Substitution", "input": "\\int\\:\\frac{\\sqrt{y^{2}-49}}{y}dy", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$" }, { "type": "step", "primary": "For $$\\sqrt{by^2-a}\\:$$substitute $$y=\\frac{\\sqrt{a}}{\\sqrt{b}}\\sec\\left(u\\right)$$<br/>$$a=49,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=7\\quad\\Rightarrow\\quad$$substitute $$y=7\\sec\\left(u\\right)$$" }, { "type": "interim", "title": "$$\\frac{dy}{du}=7\\sec\\left(u\\right)\\tan\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(7\\sec\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=7\\frac{d}{du}\\left(\\sec\\left(u\\right)\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\sec\\left(u\\right)\\right)=\\sec\\left(u\\right)\\tan\\left(u\\right)$$", "result": "=7\\sec\\left(u\\right)\\tan\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYm0apz12VWMIz5u7Qk8UfsaQp7tdIFyr1eVqMMLZHDTGohhOHWGwf8y5cOfwIkROhTAHJ4EIoVtlc6pPKQkHBcQ/y9DKGIPglJ+qMi9xDu2KqnMgpksTlT5rhoRtiJX0xDAHJ4EIoVtlc6pPKQkHBcSwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dy=7\\sec\\left(u\\right)\\tan\\left(u\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{\\sqrt{\\left(7\\sec\\left(u\\right)\\right)^{2}-49}}{7\\sec\\left(u\\right)}\\cdot\\:7\\sec\\left(u\\right)\\tan\\left(u\\right)du" }, { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{\\left(7\\sec\\left(u\\right)\\right)^{2}-49}}{7\\sec\\left(u\\right)}\\cdot\\:7\\sec\\left(u\\right)\\tan\\left(u\\right):{\\quad}7\\tan^{2}\\left(u\\right)$$", "input": "\\frac{\\sqrt{\\left(7\\sec\\left(u\\right)\\right)^{2}-49}}{7\\sec\\left(u\\right)}\\cdot\\:7\\sec\\left(u\\right)\\tan\\left(u\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{\\left(7\\sec\\left(u\\right)\\right)^{2}-49}\\cdot\\:7\\sec\\left(u\\right)\\tan\\left(u\\right)}{7\\sec\\left(u\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$7$$", "result": "=\\frac{\\sqrt{\\left(7\\sec\\left(u\\right)\\right)^{2}-49}\\sec\\left(u\\right)\\tan\\left(u\\right)}{\\sec\\left(u\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sec\\left(u\\right)$$", "result": "=\\sqrt{\\left(7\\sec\\left(u\\right)\\right)^{2}-49}\\tan\\left(u\\right)" }, { "type": "interim", "title": "$$\\sqrt{\\left(7\\sec\\left(u\\right)\\right)^{2}-49}=7\\sqrt{\\sec^{2}\\left(u\\right)-1}$$", "input": "\\sqrt{\\left(7\\sec\\left(u\\right)\\right)^{2}-49}", "steps": [ { "type": "interim", "title": "$$\\left(7\\sec\\left(u\\right)\\right)^{2}=49\\sec^{2}\\left(u\\right)$$", "input": "\\left(7\\sec\\left(u\\right)\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=7^{2}\\sec^{2}\\left(u\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$7^{2}=49$$", "result": "=49\\sec^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f4z4lXNqKLWUhOLK1EDt73yRHuGw7+tM5METTDj6vVGhz+tlIMCIW26TbuUvGCKyvrU+2UoDKiEVSa+cG/r2Cz13lZjvOze2j8VegTl5r5sIEPHwsJZX5MLrEb1MD0ppLtHscpJeQqWT9JCltj5eSw==" } }, { "type": "step", "result": "=\\sqrt{49\\sec^{2}\\left(u\\right)-49}" }, { "type": "interim", "title": "Factor $$49\\sec^{2}\\left(u\\right)-49:{\\quad}49\\left(\\sec^{2}\\left(u\\right)-1\\right)$$", "input": "49\\sec^{2}\\left(u\\right)-49", "result": "=\\sqrt{49\\left(\\sec^{2}\\left(u\\right)-1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=49\\sec^{2}\\left(u\\right)-49\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$49$$", "result": "=49\\left(\\sec^{2}\\left(u\\right)-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\sqrt{49}\\sqrt{\\sec^{2}\\left(u\\right)-1}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{49}=7$$", "input": "\\sqrt{49}", "result": "=7\\sqrt{\\sec^{2}\\left(u\\right)-1}", "steps": [ { "type": "step", "primary": "Factor the number: $$49=7^{2}$$", "result": "=\\sqrt{7^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{7^{2}}=7$$" ], "result": "=7", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vRl/EfM+TKnU4p6kBedwNdkzwQPDTrXqSlNVb9ufVNsgJ/ZZA32ZInFBpDtxBfiK3Ue7AAAHoiJINHBQTAsOlg/ugVE+X7AoCUA6pyPNDl8/y9DKGIPglJ+qMi9xDu2K7Uw14T6zAJmaTiGIxAOPo9d7olj13zQ2JpeQ2VH6yvvD7kWKpcy0ua6eqK2ZzORmbGUhD9+gDQg079F600Xu5Q==" } }, { "type": "step", "result": "=7\\tan\\left(u\\right)\\sqrt{\\sec^{2}\\left(u\\right)-1}" }, { "type": "interim", "title": "$$\\sqrt{\\sec^{2}\\left(u\\right)-1}=\\tan\\left(u\\right)$$", "input": "\\sqrt{\\sec^{2}\\left(u\\right)-1}", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\sec^{2}\\left(x\\right)=\\tan^{2}\\left(x\\right)+1$$", "secondary": [ "$$\\sec^{2}\\left(x\\right)-1=\\tan^{2}\\left(x\\right)$$" ], "result": "=\\sqrt{\\tan^{2}\\left(u\\right)}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "result": "=\\tan\\left(u\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pNgPPjxfgQQy4WKXvL0D2jxU+xCbrbDzZfDMAOs5Zsp1g99dC9fj9sg0EHzBIRDROFfWHbiVQMq7hoOw+iT3UPae3DzsVDZzEjALKmoX1uKxoUUXUzX7QSuGYiTkt+V2GO+JEu1ajQ9OVlRrNMfYTQ==" } }, { "type": "step", "result": "=7\\tan\\left(u\\right)\\tan\\left(u\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\tan\\left(u\\right)\\tan\\left(u\\right)=\\:\\tan^{1+1}\\left(u\\right)$$" ], "result": "=7\\tan^{1+1}\\left(u\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=7\\tan^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:7\\tan^{2}\\left(u\\right)du" } ], "meta": { "interimType": "Integral Trig Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7891PvlhYwmsQicv27CZKI/r+xRjfc4zPQEB3dxlCBEmSUM9pakkKILvT6Fs/PM359VEgBuKxZgOTX2ljN4yVJ9qQggBPzB4Qayiyi1+p6hLgSEHRda+G5BHM5FRE2/NG/rPtjkzrp/qXVXoSjdw9RCoGY7zz/jh+vCmMFirn8hOUgb+os4Xi1VCkP3pqJAFMvaRT1VJNfAGrb1v7ECDlIa6Czw5iyt3S4rqYGlwS4MW" } }, { "type": "step", "result": "=\\int\\:7\\tan^{2}\\left(u\\right)du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=7\\cdot\\:\\int\\:\\tan^{2}\\left(u\\right)du" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\int\\:\\tan^{2}\\left(u\\right)du", "result": "=7\\cdot\\:\\int\\:-1+\\sec^{2}\\left(u\\right)du", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\sec^{2}\\left(x\\right)-\\tan^{2}\\left(x\\right)=1$$", "secondary": [ "Therefore $$\\tan^{2}\\left(x\\right)=-1+\\sec^{2}\\left(x\\right)$$" ], "result": "=\\int\\:-1+\\sec^{2}\\left(u\\right)du" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s78WEuOhxXvvHep2+O4/Z3yb1fWutlCU4zUzgkFchUVlKOeWgsE4Mk40prEDZkKQ8Xcq64+b8YguXf4qCtKW9b7JFhxSzNcjgPzX10I7nnzbPl8aBoWkTNqzhxIOVtYbHI1xVRRqi3IfGJ/VIGue/q8rbdeLci9PfzmQBgUrYJA7EGQkYHdLAgDN98wwK2L0GGtYz5H/m/M8t6DG6EYCagD4=" } }, { "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": "=7\\left(-\\int\\:1du+\\int\\:\\sec^{2}\\left(u\\right)du\\right)" }, { "type": "interim", "title": "$$\\int\\:1du=u$$", "input": "\\int\\:1du", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:u" }, { "type": "step", "primary": "Simplify", "result": "=u", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\sec^{2}\\left(u\\right)du=\\tan\\left(u\\right)$$", "input": "\\int\\:\\sec^{2}\\left(u\\right)du", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:\\sec^{2}\\left(u\\right)du=\\tan\\left(u\\right)$$", "result": "=\\tan\\left(u\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s76WhFEDxKRsZ5Oy1LNu8NlhfRzuAubUUJYHDgzAJMdz/8+LBiPyAP34u+4MuPtfQQh2j7LTAccdz2lzIXeuICKs9qmCX2Ps8pKcZArpcqVrSYxxOM61DaQMHzXza8hWQ06tloVOYT6ozcw7EpiMvtWE=" } }, { "type": "step", "result": "=7\\left(-u+\\tan\\left(u\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\arcsec\\left(\\frac{1}{7}y\\right)$$", "result": "=7\\left(-\\arcsec\\left(\\frac{1}{7}y\\right)+\\tan\\left(\\arcsec\\left(\\frac{1}{7}y\\right)\\right)\\right)" }, { "type": "interim", "title": "Simplify $$7\\left(-\\arcsec\\left(\\frac{1}{7}y\\right)+\\tan\\left(\\arcsec\\left(\\frac{1}{7}y\\right)\\right)\\right):{\\quad}-7\\arcsec\\left(\\frac{1}{7}y\\right)+\\sqrt{y^{2}-49}$$", "input": "7\\left(-\\arcsec\\left(\\frac{1}{7}y\\right)+\\tan\\left(\\arcsec\\left(\\frac{1}{7}y\\right)\\right)\\right)", "result": "=-7\\arcsec\\left(\\frac{1}{7}y\\right)+\\sqrt{y^{2}-49}", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\tan\\left(\\arcsec\\left(x\\right)\\right)=\\sqrt{x^{2}-1}$$", "result": "=7\\left(-\\arcsec\\left(\\frac{1}{7}y\\right)+\\sqrt{\\left(\\frac{1}{7}y\\right)^{2}-1}\\right)" }, { "type": "interim", "title": "$$\\sqrt{\\left(\\frac{1}{7}y\\right)^{2}-1}=\\frac{\\sqrt{y^{2}-49}}{7}$$", "input": "\\sqrt{\\left(\\frac{1}{7}y\\right)^{2}-1}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{1}{7}y\\right)^{2}=\\frac{1}{49}y^{2}$$", "input": "\\left(\\frac{1}{7}y\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=\\left(\\frac{1}{7}\\right)^{2}y^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "secondary": [ "$$\\left(\\frac{1}{7}\\right)^{2}=\\frac{1^{2}}{7^{2}}$$" ], "result": "=\\frac{1^{2}}{7^{2}}y^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{1}{7^{2}}y^{2}" }, { "type": "step", "primary": "$$7^{2}=49$$", "result": "=\\frac{1}{49}y^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rJ1dpiFGfPlVsEVgrUsBMOiEPDD5lvIAC9CzFeUpV5JwkKGJWEPFPk38sdJMsyPIsv4rPC5XP2kBT9/dGtf3kYdEeJB8NSwK2cnf2Bc1WE3bKE5wP7sJKrpl4eFiWbAEKRVGXwWPb1Y2Gdym7I+2D+lAY90bDZpf8h0CmcgfIw0=" } }, { "type": "step", "result": "=\\sqrt{\\frac{1}{49}y^{2}-1}" }, { "type": "interim", "title": "Join $$\\frac{1}{49}y^{2}-1:{\\quad}\\frac{y^{2}-49}{49}$$", "input": "\\frac{1}{49}y^{2}-1", "result": "=\\sqrt{\\frac{y^{2}-49}{49}}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{1}{49}y^{2}\\::{\\quad}\\frac{y^{2}}{49}$$", "input": "\\frac{1}{49}y^{2}", "result": "=\\frac{y^{2}}{49}-1", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:y^{2}}{49}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:y^{2}=y^{2}$$", "result": "=\\frac{y^{2}}{49}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:49}{49}$$", "result": "=\\frac{y^{2}}{49}-\\frac{1\\cdot\\:49}{49}" }, { "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{y^{2}-1\\cdot\\:49}{49}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:49=49$$", "result": "=\\frac{y^{2}-49}{49}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$", "result": "=\\frac{\\sqrt{y^{2}-49}}{\\sqrt{49}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{49}=7$$", "input": "\\sqrt{49}", "result": "=\\frac{\\sqrt{y^{2}-49}}{7}", "steps": [ { "type": "step", "primary": "Factor the number: $$49=7^{2}$$", "result": "=\\sqrt{7^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{7^{2}}=7$$" ], "result": "=7", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78C7hkpu/Joug4mhUtfRKn5G+slUdHIYX0MV2VjqY2LctOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOE2Vfe9OTlRKexmtZToyb01J7YHBstuwBvJ7ETNaqQHLz0u0oYyhHJnT30W0t41VHKlyY7Oi7qtu1PGtZvH4RBY+rXJcv9bNO5HkWtH0Isy4JEEBbXPOpm9SyjfLzcABvrCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=7\\left(\\frac{\\sqrt{y^{2}-49}}{7}-\\arcsec\\left(\\frac{1}{7}y\\right)\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=7,\\:b=-\\arcsec\\left(\\frac{1}{7}y\\right),\\:c=\\frac{\\sqrt{y^{2}-49}}{7}$$" ], "result": "=7\\left(-\\arcsec\\left(\\frac{1}{7}y\\right)\\right)+7\\cdot\\:\\frac{\\sqrt{y^{2}-49}}{7}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-7\\arcsec\\left(\\frac{1}{7}y\\right)+7\\cdot\\:\\frac{\\sqrt{y^{2}-49}}{7}" }, { "type": "interim", "title": "$$7\\cdot\\:\\frac{\\sqrt{y^{2}-49}}{7}=\\sqrt{y^{2}-49}$$", "input": "7\\cdot\\:\\frac{\\sqrt{y^{2}-49}}{7}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{y^{2}-49}\\cdot\\:7}{7}" }, { "type": "step", "primary": "Cancel the common factor: $$7$$", "result": "=\\sqrt{y^{2}-49}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7uxjTDNyZVStXsVC0PezoqXkXb/ZA2jW74r1Uru/ZjO/PSC9Tko2tzCofXeBadB/ycJChiVhDxT5N/LHSTLMjyGO2oX3BlHVUhjqPmiv7o0rWxXslZzdJTndHXRM4AvH+gMO+EVSAWk6FHTtJie2qQnkXb/ZA2jW74r1Uru/ZjO/X3x8QstGT1zNhEKi8Ndf6+yEacguhzsAHO2XXoHoSUw==" } }, { "type": "step", "result": "=-7\\arcsec\\left(\\frac{1}{7}y\\right)+\\sqrt{y^{2}-49}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Pzxt5GptGYo8zkM0yTUtMlzL/gAXciASkoF83BDKBQl13ezsPG+A8XSDlR4LkbA4d5UbdVPByqxMb2bCAxDiLSAn9lkDfZkicUGkO3EF+Ip86JzxTterJJ5TnvuzGWM65PdBU3AgUw6DmAhrUOsknoBSbhj2tClc4BF1ITPUabu5yWYJ+SIE4qbz8ogRifzl7kAjP76qW66lOUsURwT0nSGBZ0dMt49amvbYBSw7UW2dvEeSEA4NQRxr08Pvey9kcaaKD+l4TLTxn5tK6GLQgltUmi98REVBrV8MWoqMWG2wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-7\\arcsec\\left(\\frac{1}{7}y\\right)+\\sqrt{y^{2}-49}+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=Trig%20Power%20Multiplication", "practiceTopic": "Integral Trig Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "y", "plotRequest": "y=-7\\arcsec(\\frac{1}{7}y)+\\sqrt{y^{2}-49}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }