integral of (-x+2)/(x^2-x+1)

{ "query": { "display": "$$\\int\\:\\frac{-x+2}{x^{2}-x+1}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{-x+2}{x^{2}-x+1}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{1}{2}\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|+\\sqrt{3}\\arctan(\\frac{1}{\\sqrt{3}}(2x-1))+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{-x+2}{x^{2}-x+1}dx=-\\frac{1}{2}\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|+\\sqrt{3}\\arctan\\left(\\frac{1}{\\sqrt{3}}\\left(2x-1\\right)\\right)+C$$", "input": "\\int\\:\\frac{-x+2}{x^{2}-x+1}dx", "steps": [ { "type": "interim", "title": "Complete the square $$x^{2}-x+1:{\\quad}\\left(x-\\frac{1}{2}\\right)^{2}+\\frac{3}{4}$$", "input": "x^{2}-x+1", "steps": [ { "type": "step", "primary": "Write $$x^{2}-x+1\\:$$in the form: $$x^2+2ax+a^2$$" }, { "type": "interim", "title": "$$2a=-1{\\quad:\\quad}a=-\\frac{1}{2}$$", "input": "2a=-1", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=-1", "result": "a=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{-1}{2}" }, { "type": "step", "primary": "Simplify", "result": "a=-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-\\frac{1}{2}\\right)^{2}\\:$$", "result": "=x^{2}-x+1+\\left(-\\frac{1}{2}\\right)^{2}-\\left(-\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}-1x+\\left(-\\frac{1}{2}\\right)^{2}=\\left(x-\\frac{1}{2}\\right)^{2}$$", "Complete the square" ], "result": "=\\left(x-\\frac{1}{2}\\right)^{2}+1-\\left(-\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "=\\left(x-\\frac{1}{2}\\right)^{2}+\\frac{3}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{-x+2}{\\left(x-\\frac{1}{2}\\right)^{2}+\\frac{3}{4}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{-x+2}{\\left(x-\\frac{1}{2}\\right)^{2}+\\frac{3}{4}}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=x-\\frac{1}{2}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=1$$", "input": "\\frac{d}{dx}\\left(x-\\frac{1}{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(\\frac{1}{2}\\right)" }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(\\frac{1}{2}\\right)=0$$", "input": "\\frac{d}{dx}\\left(\\frac{1}{2}\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg0Q/RY4sqo8t3cGZGHqcAt44OmnsvrgbNkOIEFbFW8WYAIXwDG7/aP+CNC9gUVzaHNS9SX5M3gDB/Er/MAH1V+5QV7agSZLIzF7D9vX0CHvIkpfTXQGxfmhHXAARBcv67CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=1dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=1du$$" }, { "type": "step", "result": "=\\int\\:\\frac{-x+2}{u^{2}+\\frac{3}{4}}\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:\\frac{4\\left(-x+2\\right)}{4u^{2}+3}du" }, { "type": "interim", "title": "$$u=x-\\frac{1}{2}\\quad\\Rightarrow\\quad\\:x=u+\\frac{1}{2}$$", "input": "x-\\frac{1}{2}=u", "steps": [ { "type": "interim", "title": "Move $$\\frac{1}{2}\\:$$to the right side", "input": "x-\\frac{1}{2}=u", "result": "x=u+\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Add $$\\frac{1}{2}$$ to both sides", "result": "x-\\frac{1}{2}+\\frac{1}{2}=u+\\frac{1}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=u+\\frac{1}{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\frac{4\\left(-\\left(u+\\frac{1}{2}\\right)+2\\right)}{4u^{2}+3}du" }, { "type": "interim", "title": "Simplify $$\\frac{4\\left(-\\left(u+\\frac{1}{2}\\right)+2\\right)}{4u^{2}+3}:{\\quad}\\frac{2\\left(-2u+3\\right)}{4u^{2}+3}$$", "input": "\\frac{4\\left(-\\left(u+\\frac{1}{2}\\right)+2\\right)}{4u^{2}+3}", "steps": [ { "type": "interim", "title": "$$-\\left(u+\\frac{1}{2}\\right):{\\quad}-u-\\frac{1}{2}$$", "input": "-\\left(u+\\frac{1}{2}\\right)", "result": "=\\frac{4\\left(-u+2-\\frac{1}{2}\\right)}{4u^{2}+3}", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(u\\right)-\\left(\\frac{1}{2}\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-u-\\frac{1}{2}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Join $$-u-\\frac{1}{2}+2:{\\quad}\\frac{-2u+3}{2}$$", "input": "-u-\\frac{1}{2}+2", "result": "=\\frac{4\\cdot\\:\\frac{-2u+3}{2}}{4u^{2}+3}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$u=\\frac{u2}{2},\\:2=\\frac{2\\cdot\\:2}{2}$$", "result": "=-\\frac{u\\cdot\\:2}{2}-\\frac{1}{2}+\\frac{2\\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{-u\\cdot\\:2-1+2\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$-u\\cdot\\:2-1+2\\cdot\\:2=-2u+3$$", "input": "-u\\cdot\\:2-1+2\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=-2u-1+4" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+4=3$$", "result": "=-2u+3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7htil2SJjKRK4jtARhHtgL4nDdKucBi4LmQUhhh+vc191g99dC9fj9sg0EHzBIRDRMQ/uQM4GVcUtiNLKzTXIJ4t933es/tHqOEyJb81AzWnxsn+BgPyPcUvOeOw0iOSbCRbQDU3R08K4waZEMYB8xw==" } }, { "type": "step", "result": "=\\frac{-2u+3}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "Multiply $$4\\cdot\\:\\frac{-2u+3}{2}\\::{\\quad}2\\left(-2u+3\\right)$$", "input": "4\\cdot\\:\\frac{-2u+3}{2}", "result": "=\\frac{2\\left(-2u+3\\right)}{4u^{2}+3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(-2u+3\\right)\\cdot\\:4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2\\left(-2u+3\\right)" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{2\\left(-2u+3\\right)}{4u^{2}+3}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s79JBkXsUIjy5iAyA3iKIIfyFzaRZb2HAY+qB8Z1EcUIiM35HWg0b6UbXll/dOS8YwRC1Hih0bIV8/5v07jR9hMzsKa5Ug/T8MyUW3rAT2f4CkIkO3JmOEfVubFU6QQCNj2+NfIFBFNfsb20MbrTUbNNzwoKHQQ82OZKNTFOpAgjhizP0HeLSvgH2Txf3lKxtpU3kCh3oevUunZ7/b0qFKBQtpKudeWylWooM2AOjV8O20Puo9TbuECACF4YO4X20KQ==" } }, { "type": "step", "result": "=\\int\\:\\frac{2\\left(-2u+3\\right)}{4u^{2}+3}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": "=2\\cdot\\:\\int\\:\\frac{-2u+3}{4u^{2}+3}du" }, { "type": "interim", "title": "Apply Integral Substitution", "input": "\\int\\:\\frac{-2u+3}{4u^{2}+3}du", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=\\frac{\\sqrt{3}}{2}v$$" ] }, { "type": "step", "primary": "For $$bx^2\\pm\\:a\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}u$$<br/>$$a=3,\\:b=4,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=\\frac{\\sqrt{3}}{2}\\quad\\Rightarrow\\quad$$substitute $$x=\\frac{\\sqrt{3}}{2}u$$" }, { "type": "interim", "title": "$$\\frac{du}{dv}=\\frac{\\sqrt{3}}{2}$$", "input": "\\frac{d}{dv}\\left(\\frac{\\sqrt{3}}{2}v\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\frac{\\sqrt{3}}{2}\\frac{dv}{dv}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dv}{dv}=1$$", "result": "=\\frac{\\sqrt{3}}{2}\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{\\sqrt{3}}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuveuP9TvxI8xfexNOJkn2uRVQx6OIaycFcyk7LKy6m3pN4cZPWgnwFqHQUcV4FHsdMQ0vmOWo9IZfhei7w2gJYfy7/etscpfymSU8nOLWbWZEt3ZXAiqUE0HIXrrrezJJS6lcBkKYvNvx/R4XorlAqDz7IqY/3J3qrwMvKWPYN+jwE87HTCWyAU3ypRroDMDQ==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{\\sqrt{3}}{2}dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{-2\\cdot\\:\\frac{\\sqrt{3}}{2}v+3}{4\\left(\\frac{\\sqrt{3}}{2}v\\right)^{2}+3}\\cdot\\:\\frac{\\sqrt{3}}{2}dv" }, { "type": "interim", "title": "Simplify $$\\frac{-2\\cdot\\:\\frac{\\sqrt{3}}{2}v+3}{4\\left(\\frac{\\sqrt{3}}{2}v\\right)^{2}+3}\\cdot\\:\\frac{\\sqrt{3}}{2}:{\\quad}\\frac{-v+\\sqrt{3}}{2\\left(v^{2}+1\\right)}$$", "input": "\\frac{-2\\cdot\\:\\frac{\\sqrt{3}}{2}v+3}{4\\left(\\frac{\\sqrt{3}}{2}v\\right)^{2}+3}\\cdot\\:\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{-2\\cdot\\:\\frac{\\sqrt{3}}{2}v+3}{4\\left(\\frac{\\sqrt{3}}{2}v\\right)^{2}+3}=\\frac{-v+\\sqrt{3}}{\\sqrt{3}\\left(v^{2}+1\\right)}$$", "input": "\\frac{-2\\cdot\\:\\frac{\\sqrt{3}}{2}v+3}{4\\left(\\frac{\\sqrt{3}}{2}v\\right)^{2}+3}", "steps": [ { "type": "interim", "title": "$$4\\left(\\frac{\\sqrt{3}}{2}v\\right)^{2}=3v^{2}$$", "input": "4\\left(\\frac{\\sqrt{3}}{2}v\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{\\sqrt{3}}{2}v\\right)^{2}=\\frac{3v^{2}}{2^{2}}$$", "input": "\\left(\\frac{\\sqrt{3}}{2}v\\right)^{2}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{\\sqrt{3}}{2}v\\::{\\quad}\\frac{\\sqrt{3}v}{2}$$", "input": "\\frac{\\sqrt{3}}{2}v", "result": "=\\left(\\frac{\\sqrt{3}v}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{3}v}{2}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(\\sqrt{3}v\\right)^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "secondary": [ "$$\\left(\\sqrt{3}v\\right)^{2}=\\left(\\sqrt{3}\\right)^{2}v^{2}$$" ], "result": "=\\frac{\\left(\\sqrt{3}\\right)^{2}v^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{3v^{2}}{2^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7W+TF9DwfoU38VHY68TaD4x3iuF/SPZhI7N4jfvnnOBXehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpg0n3lV8GInwArTxEeycUbnxKuqsNbiYVwQr6E82pk3yJv3vPhyJEHOBe1ydiXxyKeUZ0FwFLOHJOQF8OMTXcTu82/iYutSkun5VQLLyskJisIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=4\\cdot\\:\\frac{3v^{2}}{2^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3v^{2}\\cdot\\:4}{2^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:4=12$$", "result": "=\\frac{12v^{2}}{2^{2}}" }, { "type": "interim", "title": "Factor $$12:{\\quad}2^{2}\\cdot\\:3$$", "steps": [ { "type": "step", "primary": "Factor $$12=2^{2}\\cdot\\:3$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{2^{2}\\cdot\\:3v^{2}}{2^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$2^{2}$$", "result": "=3v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LG24uscxBAhZDGMIKyI10oBgtMsPjEIiAcn2BQSfrlNV00rpv8+ZC6TM10tVCSHsYAsXL0SggoaWzn1E3qRqh/8//6/nV5O4fb8Xgwi7maoe5Z1qy8+tbSMawVSK21U7aQpBvu/f/WjXJTGpV8NfqMXcN6cRaiMtIZdgwNwCXts=" } }, { "type": "step", "result": "=\\frac{-2\\cdot\\:\\frac{\\sqrt{3}}{2}v+3}{3v^{2}+3}" }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{\\sqrt{3}}{2}v=\\sqrt{3}v$$", "input": "2\\cdot\\:\\frac{\\sqrt{3}}{2}v", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{3}\\cdot\\:2v}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\sqrt{3}v" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviqEAjwDQe7c0ChTseSfAE0ktOtZYwUjyXhDTsNnn6Elr3q98Jf9iPRiaoGzcVEGGEiP2VN9dPczdxV+wFkDgANdAurlQ4emtsvZr/Cm3cGdYg8+yKmP9yd6q8DLylj2DflBpwzHY/l87FbyiR1ieGZwkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\frac{-\\sqrt{3}v+3}{3v^{2}+3}" }, { "type": "interim", "title": "Factor $$-\\sqrt{3}v+3:{\\quad}\\sqrt{3}\\left(-v+\\sqrt{3}\\right)$$", "input": "-\\sqrt{3}v+3", "result": "=\\frac{\\sqrt{3}\\left(-v+\\sqrt{3}\\right)}{3v^{2}+3}", "steps": [ { "type": "step", "primary": "$$3=\\sqrt{3}\\sqrt{3}$$", "result": "=-\\sqrt{3}v+\\sqrt{3}\\sqrt{3}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$3^{2\\cdot\\:\\frac{1}{2}}=\\sqrt{3}\\sqrt{3}$$" ], "result": "=-\\sqrt{3}v+\\sqrt{3}\\sqrt{3}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$\\sqrt{3}$$", "result": "=\\sqrt{3}\\left(-v+\\sqrt{3}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$3v^{2}+3:{\\quad}3\\left(v^{2}+1\\right)$$", "input": "3v^{2}+3", "result": "=\\frac{\\sqrt{3}\\left(-v+\\sqrt{3}\\right)}{3\\left(v^{2}+1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=3v^{2}+3\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$3$$", "result": "=3\\left(v^{2}+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{3}\\left(-v+\\sqrt{3}\\right)}{3\\left(v^{2}+1\\right)}:{\\quad}\\frac{-v+\\sqrt{3}}{\\sqrt{3}\\left(v^{2}+1\\right)}$$", "input": "\\frac{\\sqrt{3}\\left(-v+\\sqrt{3}\\right)}{3\\left(v^{2}+1\\right)}", "result": "=\\frac{-v+\\sqrt{3}}{\\sqrt{3}\\left(v^{2}+1\\right)}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{3}=3^{\\frac{1}{2}}$$" ], "result": "=\\frac{3^{\\frac{1}{2}}\\left(-v+\\sqrt{3}\\right)}{3\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{3^{\\frac{1}{2}}}{3^{1}}=\\frac{1}{3^{1-\\frac{1}{2}}}$$" ], "result": "=\\frac{-v+\\sqrt{3}}{3^{-\\frac{1}{2}+1}\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$1-\\frac{1}{2}=\\frac{1}{2}$$", "result": "=\\frac{-v+\\sqrt{3}}{3^{\\frac{1}{2}}\\left(v^{2}+1\\right)}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "secondary": [ "$$3^{\\frac{1}{2}}=\\sqrt{3}$$" ], "result": "=\\frac{-v+\\sqrt{3}}{\\sqrt{3}\\left(v^{2}+1\\right)}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjGbDBFglrpcaegmxDUIxGkrDEovKhCLwQcaGrhGtGjNAKJYch51iAlBZF5PTnAR23CQoYlYQ8U+Tfyx0kyzI8iSsunKrzErbNhNv1ILsrj9DmzYFX2lVkzDUcS97LGGP5tk5tZX7n3ZiyD+OYYCxpZ6pfF1z6umzUJTJvt+ojYZ3Rhao/hEKgGYcV+JSwxeA/rqMsLqK/dc9qxAg4pAGjf3FXBkkuYJW0ShHkDxj47mEmcRvx50wuMYcUqA2dQQYQ==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75IEecii9UBK9d3BhkqbLna3Yn1NFotSssbSwtvx+GFZZQMXVQROEbsCq7cG1F2dgJyDS30q59Dn8RzHHkwHvjUGNR+nLzczAvQEfzvH/gZ/NGoPE9TME3q+OPmgkv2RQfrwqTJx6CPaGFr/18epi8V3i4MszkJdhaGh8t5BgFqhNNQ7jYuX5AKQ8XYyXATZDZEt3ZXAiqUE0HIXrrrezJB76AlcDlUi1L1rSrrxDkEzGxhEJDVFUpr44MofH32aQhHw9xx0FRW+IcIASa9xO+9ffNMXtWA3/RfrsyYeXQ5bD88cBY3bnIaVzB6efs/UiEBykTxIaGd1CB3kcAOG0ZlOgacGCDtIN17VGIABTZOKwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{-v+\\sqrt{3}}{\\sqrt{3}\\left(v^{2}+1\\right)}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{\\left(-v+\\sqrt{3}\\right)\\sqrt{3}}{\\sqrt{3}\\left(v^{2}+1\\right)\\cdot\\:2}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "result": "=\\frac{-v+\\sqrt{3}}{\\left(v^{2}+1\\right)\\cdot\\:2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{-v+\\sqrt{3}}{2\\left(v^{2}+1\\right)}dv" } ], "meta": { "interimType": "Integral Substitution 1Eq" } }, { "type": "step", "result": "=2\\cdot\\:\\int\\:\\frac{-v+\\sqrt{3}}{2\\left(v^{2}+1\\right)}dv" }, { "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\\:\\frac{1}{2}\\cdot\\:\\int\\:\\frac{-v+\\sqrt{3}}{v^{2}+1}dv" }, { "type": "interim", "title": "Expand $$\\frac{-v+\\sqrt{3}}{v^{2}+1}:{\\quad}-\\frac{v}{v^{2}+1}+\\frac{\\sqrt{3}}{v^{2}+1}$$", "input": "\\frac{-v+\\sqrt{3}}{v^{2}+1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "result": "=-\\frac{v}{v^{2}+1}+\\frac{\\sqrt{3}}{v^{2}+1}" } ], "meta": { "interimType": "Algebraic Manipulation Expand 1Eq" } }, { "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": "=2\\cdot\\:\\frac{1}{2}\\left(-\\int\\:\\frac{v}{v^{2}+1}dv+\\int\\:\\frac{\\sqrt{3}}{v^{2}+1}dv\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{v}{v^{2}+1}dv=\\frac{1}{2}\\ln\\left|v^{2}+1\\right|$$", "input": "\\int\\:\\frac{v}{v^{2}+1}dv", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{v}{v^{2}+1}dv", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$w=v^{2}+1$$" ] }, { "type": "interim", "title": "$$\\frac{dw}{dv}=2v$$", "input": "\\frac{d}{dv}\\left(v^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dv}\\left(v^{2}\\right)+\\frac{d}{dv}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(v^{2}\\right)=2v$$", "input": "\\frac{d}{dv}\\left(v^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2v^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiHI6i/lNYJouYKcylLF6B2k3hxk9aCfAWodBRxXgUexsoRboyLdWbDdDvojbJb4SkeCBKuYKgaNJ253gLI69U5feCPJC8Uak4mwlsl/8zOjPWUEL+I3n8Z72JloyPMrWQ==" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(1\\right)=0$$", "input": "\\frac{d}{dv}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYrYlw0sQkDeBfYmbOrQ+t91J8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtHFi1sZ1xQhHh3bCtbuI/B" } }, { "type": "step", "result": "=2v+0" }, { "type": "step", "primary": "Simplify", "result": "=2v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dw=2vdv$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=\\frac{1}{2v}dw$$" }, { "type": "step", "result": "=\\int\\:\\frac{v}{w}\\cdot\\:\\frac{1}{2v}dw" }, { "type": "interim", "title": "Simplify $$\\frac{v}{w}\\cdot\\:\\frac{1}{2v}:{\\quad}\\frac{1}{2w}$$", "input": "\\frac{v}{w}\\cdot\\:\\frac{1}{2v}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{v\\cdot\\:1}{w\\cdot\\:2v}" }, { "type": "step", "primary": "Cancel the common factor: $$v$$", "result": "=\\frac{1}{w\\cdot\\:2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{2w}dw" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74LieaeH1AQvqWM9xC078bzMdHbHyNly07oC4m7we4Kc2NCwimM7dB8C524HvMCB4ymZOc9q9xxqJAg2jt99wha4eveJYYFfQMHyfOzMWdYh6DAXZhJ0XdOjhz/T74T8n0UqTd96MWTKI6Kr2Ib0iQBZegS2gwh8pq/gwNfwaDRbAwNT33I9ftOGdlyhcHnXlA==" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{2w}dw" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{2}\\cdot\\:\\int\\:\\frac{1}{w}dw" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{w}dw=\\ln\\left(\\left|w\\right|\\right)$$", "result": "=\\frac{1}{2}\\ln\\left|w\\right|" }, { "type": "step", "primary": "Substitute back $$w=v^{2}+1$$", "result": "=\\frac{1}{2}\\ln\\left|v^{2}+1\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{\\sqrt{3}}{v^{2}+1}dv=\\sqrt{3}\\arctan\\left(v\\right)$$", "input": "\\int\\:\\frac{\\sqrt{3}}{v^{2}+1}dv", "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": "=\\sqrt{3}\\cdot\\:\\int\\:\\frac{1}{v^{2}+1}dv" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{v^{2}+1}dv=\\arctan\\left(v\\right)$$", "result": "=\\sqrt{3}\\arctan\\left(v\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|v^{2}+1\\right|+\\sqrt{3}\\arctan\\left(v\\right)\\right)" }, { "type": "interim", "title": "Substitute back", "input": "2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|v^{2}+1\\right|+\\sqrt{3}\\arctan\\left(v\\right)\\right)", "result": "=2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|+\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "Substitute back $$v=\\frac{2}{\\sqrt{3}}u$$", "result": "=2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}u\\right)^{2}+1\\right|+\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}u\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=x-\\frac{1}{2}$$", "result": "=2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|+\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)\\right)" } ], "meta": { "interimType": "Generic Substitute Back 0Eq" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|+\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)\\right):{\\quad}-\\frac{1}{2}\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|+\\sqrt{3}\\arctan\\left(\\frac{1}{\\sqrt{3}}\\left(2x-1\\right)\\right)$$", "input": "2\\cdot\\:\\frac{1}{2}\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|+\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)\\right)", "result": "=-\\frac{1}{2}\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|+\\sqrt{3}\\arctan\\left(\\frac{1}{\\sqrt{3}}\\left(2x-1\\right)\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|+\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)\\right)}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1\\cdot\\:\\left(-\\frac{1}{2}\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|+\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)\\right)" }, { "type": "interim", "title": "$$\\frac{1}{2}\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|=\\frac{\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|}{2}$$", "input": "\\frac{1}{2}\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|}{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|=\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|$$", "result": "=\\frac{\\ln\\left|\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}+1\\right|}{2}" }, { "type": "interim", "title": "$$\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}=\\frac{4}{3}\\left(x^{2}-x+\\frac{1}{4}\\right)$$", "input": "\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=\\left(\\frac{2}{\\sqrt{3}}\\right)^{2}\\left(x-\\frac{1}{2}\\right)^{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": [ "$$\\left(\\frac{2}{\\sqrt{3}}\\right)^{2}=\\frac{2^{2}}{\\left(\\sqrt{3}\\right)^{2}}$$" ], "result": "=\\frac{2^{2}}{\\left(\\sqrt{3}\\right)^{2}}\\left(x-\\frac{1}{2}\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{2^{2}}{3}\\left(x-\\frac{1}{2}\\right)^{2}" }, { "type": "interim", "title": "$$\\left(x-\\frac{1}{2}\\right)^{2}=x^{2}-x+\\frac{1}{4}$$", "input": "\\left(x-\\frac{1}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=x,\\:\\:b=\\frac{1}{2}$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=x^{2}-2x\\frac{1}{2}+\\left(\\frac{1}{2}\\right)^{2}" }, { "type": "interim", "title": "Simplify $$x^{2}-2x\\frac{1}{2}+\\left(\\frac{1}{2}\\right)^{2}:{\\quad}x^{2}-x+\\frac{1}{4}$$", "input": "x^{2}-2x\\frac{1}{2}+\\left(\\frac{1}{2}\\right)^{2}", "result": "=x^{2}-x+\\frac{1}{4}", "steps": [ { "type": "interim", "title": "$$2x\\frac{1}{2}=x$$", "input": "2x\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}x" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=x\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$x\\cdot\\:1=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PowF26I9cpMXCar1EVkLqgCWKUbvV6WK3fDUgFtg3Q84452XiK4JmjK/ZyKR8NAKdo32iBGpFtgPE3KhTebUJAV/rcH2AWDxUmpFfAJwUygkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\left(\\frac{1}{2}\\right)^{2}=\\frac{1}{4}$$", "input": "\\left(\\frac{1}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{1^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{1}{2^{2}}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HHy18EzOxpo1Fpth3bslTY5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdjqPCSP1OLsLEy6TGAzXCvnVaiLgjmyMQYlA0xnylLMSfAPPWAt/OA22K6WveyBN9q1zmgcD+ls8u2Jqrz0f4YEg==" } }, { "type": "step", "result": "=x^{2}-x+\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wDWsg13uDGUBLkxj63a/2UBgnzAZkUy+vMtgHHTjQTCrju+5Z51e/ZZSD3gRHwjBSEsuqTsJDSdJtueooy/8/Q/mU/6hTfHVOodxq1kvbMy1E6JRggk4Dx+L2e7PzZrH2hqXeD7QXqaM9f2BTNOW9vVI+oyuKRZpXatkOuh/SS6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{2^{2}}{3}\\left(x^{2}-x+\\frac{1}{4}\\right)" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{4}{3}\\left(x^{2}-x+\\frac{1}{4}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qyW2P31HNrBUXtu5v5OAaNZvtJeh37VUgkLUXmbijVtrVHdi1jNXT1/5ITmfawY6ICf2WQN9mSJxQaQ7cQX4il2BgpogRffZXrJl2fYiHyeVzcX/V4A/XzRgVQS94ejGeCGl/DazwvvsZZP5Fs790cy82qTZUEi5vCQD0nP8naakCTZdr9qqhEPzl2Eb4+jRQU0awAPebrVvabFrIWozM61iiafZvAhoxg++twtNO7k3qrIrHcL/2jPvZ48CJr2CCsntq/vrZXuUqRSeixMCOg==" } }, { "type": "step", "result": "=\\frac{\\ln\\left|\\frac{4}{3}\\left(x^{2}-x+\\frac{1}{4}\\right)+1\\right|}{2}" }, { "type": "interim", "title": "Expand $$\\frac{4}{3}\\left(x^{2}-x+\\frac{1}{4}\\right)+1:{\\quad}\\frac{4}{3}x^{2}-\\frac{4}{3}x+\\frac{1}{3}+1$$", "input": "\\frac{4}{3}\\left(x^{2}-x+\\frac{1}{4}\\right)+1", "result": "=\\frac{\\ln\\left|\\frac{4}{3}x^{2}-\\frac{4}{3}x+\\frac{1}{3}+1\\right|}{2}", "steps": [ { "type": "interim", "title": "Expand $$\\frac{4}{3}\\left(x^{2}-x+\\frac{1}{4}\\right):{\\quad}\\frac{4}{3}x^{2}-\\frac{4}{3}x+\\frac{1}{3}$$", "input": "\\frac{4}{3}\\left(x^{2}-x+\\frac{1}{4}\\right)", "result": "=\\frac{4}{3}x^{2}-\\frac{4}{3}x+\\frac{1}{3}+1", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=\\frac{4}{3}x^{2}+\\frac{4}{3}\\left(-x\\right)+\\frac{4}{3}\\cdot\\:\\frac{1}{4}", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=\\frac{4}{3}x^{2}-\\frac{4}{3}x+\\frac{4}{3}\\cdot\\:\\frac{1}{4}" }, { "type": "interim", "title": "Multiply $$\\frac{4}{3}\\cdot\\:\\frac{1}{4}\\::{\\quad}\\frac{1}{3}$$", "input": "\\frac{4}{3}\\cdot\\:\\frac{1}{4}", "steps": [ { "type": "step", "primary": "Cross-cancel: $$4$$", "result": "=\\frac{1}{3}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\frac{4}{3}x^{2}-\\frac{4}{3}x+\\frac{1}{3}" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UicEPR+dNEbD9r+JxZPfl68c9ai2MNBUN6Jfa19fc0r1iqBW3g42/UIzi3DHfjuVzMFYmi1F5Hg/ibpEToVnYw9A5Q0GYvIbaXvQip1JnHqPXbw0yt/HvgP5OwpXLEQxmswFZhlLP8wxASXX28CFwEUqTd96MWTKI6Kr2Ib0iQAKeq+fnnqmKwOkTB0rBQ9oIjVnE7R7/mAwmHWKic90RSHyrkM3SklQFIlUS/SrL2GwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UicEPR+dNEbD9r+JxZPfl68c9ai2MNBUN6Jfa19fc0rJy6RsDFLXEJ7h4euTlxGOq47vuWedXv2WUg94ER8IwXZ+7cf1+YsRTaPa7cnp7un9GQCIjXUaY/oOI3L27Ksua4snavGk3507GSTCuAhKKjq+Ng6EWuJZr7Vvl0wY8in8bYA0b6V2RSTOZ7Os9NODD+DgXh/k3imNkQWKACLTMq8c9ai2MNBUN6Jfa19fc0oIzwF3hYkDKwG/MMnshoxD" } }, { "type": "interim", "title": "Join $$\\frac{4}{3}x^{2}-\\frac{4}{3}x+\\frac{1}{3}+1:{\\quad}\\frac{4}{3}+\\frac{4x^{2}-4x}{3}$$", "input": "\\frac{4}{3}x^{2}-\\frac{4}{3}x+\\frac{1}{3}+1", "result": "=\\frac{\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|}{2}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{4}{3}x^{2}\\::{\\quad}\\frac{4x^{2}}{3}$$", "input": "\\frac{4}{3}x^{2}", "result": "=\\frac{4x^{2}}{3}-\\frac{4}{3}x+\\frac{1}{3}+1", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{4x^{2}}{3}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "interim", "title": "Multiply $$\\frac{4}{3}x\\::{\\quad}\\frac{4x}{3}$$", "input": "\\frac{4}{3}x", "result": "=\\frac{4x^{2}}{3}-\\frac{4x}{3}+\\frac{1}{3}+1", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{4x}{3}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "interim", "title": "Combine the fractions $$1+\\frac{1}{3}:{\\quad}\\frac{4}{3}$$", "input": "1+\\frac{1}{3}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=\\frac{1\\cdot\\:3}{3}+\\frac{1}{3}" }, { "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\\cdot\\:3+1}{3}" }, { "type": "interim", "title": "$$1\\cdot\\:3+1=4$$", "input": "1\\cdot\\:3+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=3+1" }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Q2oM7xSFGyeNpdYXLohAbd6GQqufR6tr2vPxOUv7H++Bt2nC2SiW/BhWFReLmsbAP/n/sT8Hudl/0KJRqY9qeZf/cu0bXtJSWO5jc5h9tW0=" } }, { "type": "step", "result": "=\\frac{4}{3}" } ], "meta": { "interimType": "LCD Top 1Eq" } }, { "type": "interim", "title": "Combine the fractions $$\\frac{4x^{2}}{3}-\\frac{4x}{3}:{\\quad}\\frac{4x^{2}-4x}{3}$$", "input": "\\frac{4x^{2}}{3}-\\frac{4x}{3}", "steps": [ { "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{4x^{2}-4x}{3}" } ], "meta": { "interimType": "LCD Top 1Eq" } }, { "type": "step", "result": "=\\frac{\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8SS44GUu6WQp+1vyTJ7je7CSmxGhCFuclcQhCJWZVwBt+po+vxwUHKrXeb/LPYqUxkzChY/lQ0byleqe37V/SyTehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpoLeycgw6B/tDi5B/+nN/Hq9S/IgOSh0/SsdtKAKlYwgpkYE4qfD7XeMk/rlw/SOA4OXJH7ETNpPC1JA2YIi+BVTW26qciuyUBGXQExCUedYVAhy1NZsn6qXb31QuTAD6Wklm5GeBeERo3S5+QgUMeHf5ASIvyaiWRivMjsLv3gTJwYxOF+sSyfn5xSKBuiA3CmJHQB/G3RlogCrEgMvKcnTZfZewL2G6kwUFxuSB2QzjgBE5QMI9yXIVBp6405pgf231Rc3daMG4DDCH2fNvppurCc0v6+SezXpJu+Sddf4" } }, { "type": "interim", "title": "$$\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)=\\sqrt{3}\\arctan\\left(\\frac{2x-1}{\\sqrt{3}}\\right)$$", "input": "\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right):{\\quad}\\frac{2}{\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}$$", "input": "\\frac{2}{\\sqrt{3}}\\left(x-\\frac{1}{2}\\right)", "result": "=\\sqrt{3}\\arctan\\left(\\frac{2}{\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=\\frac{2}{\\sqrt{3}},\\:b=x,\\:c=\\frac{1}{2}$$" ], "result": "=\\frac{2}{\\sqrt{3}}x-\\frac{2}{\\sqrt{3}}\\cdot\\:\\frac{1}{2}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=\\frac{2}{\\sqrt{3}}x-\\frac{1}{2}\\cdot\\:\\frac{2}{\\sqrt{3}}" }, { "type": "interim", "title": "Multiply $$\\frac{1}{2}\\cdot\\:\\frac{2}{\\sqrt{3}}\\::{\\quad}\\frac{1}{\\sqrt{3}}$$", "input": "\\frac{1}{2}\\cdot\\:\\frac{2}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Cross-cancel: $$2$$", "result": "=\\frac{1}{\\sqrt{3}}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\frac{2}{\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UuvTuPzfuNrnsZkujVu2Cs6WQrqFkMuuQuVH/qj6Fw7Rtf7VmnaSiEsGSSUZLMUmq47vuWedXv2WUg94ER8Iwbufkmzwib9pqPA2eECfrNeMN+JAd688fEnZoJAu2nSip9n3nv24e4ilT1H/IHN1YUUqTd96MWTKI6Kr2Ib0iQAKeq+fnnqmKwOkTB0rBQ9oQZNzUxMs0AYUhABKIVJ+VWebD4LXj1YMEuPUExJDomq/vgZCItqmTYjvJkSs6Kov" } }, { "type": "interim", "title": "Join $$\\frac{2}{\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}:{\\quad}\\frac{2x-1}{\\sqrt{3}}$$", "input": "\\frac{2}{\\sqrt{3}}x-\\frac{1}{\\sqrt{3}}", "result": "=\\sqrt{3}\\arctan\\left(\\frac{2x-1}{\\sqrt{3}}\\right)", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{2}{\\sqrt{3}}x\\::{\\quad}\\frac{2x}{\\sqrt{3}}$$", "input": "\\frac{2}{\\sqrt{3}}x", "result": "=\\frac{2x}{\\sqrt{3}}-\\frac{1}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2x}{\\sqrt{3}}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "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{2x-1}{\\sqrt{3}}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Qbs/yK1fqOkLwCwQf3MpEy46HDzovEEkVADEra8gd3SM/89dHJTnYq3b4cbc73nde4zHUdgwn7a39DW3xcDRjQOfOVs9mPIqDLV5QIWwt3nABwbo9SnfC1iRSaqJMM5SsbLFbH5F+2xmd45TPHFiXCDPZrupBBx68YBG4MXbevE/y9DKGIPglJ+qMi9xDu2KKaJ53RhQ2KevWytvodJNoxoagi4qxzzMSQTuROfwam7OlkK6hZDLrkLlR/6o+hcOMb3sv0wOeh8r8pjAySMiE7l0leXxZ+4tr1eU3oO4fSe7UPIOCpHpx8wpMGc5DVo9v74GQiLapk2I7yZErOiqLw==" } }, { "type": "step", "result": "=1\\cdot\\:\\left(\\sqrt{3}\\arctan\\left(\\frac{2x-1}{\\sqrt{3}}\\right)-\\frac{\\ln\\left|\\frac{4x^{2}-4x}{3}+\\frac{4}{3}\\right|}{2}\\right)" }, { "type": "step", "primary": "Refine", "result": "=-\\frac{\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|}{2}+\\sqrt{3}\\arctan\\left(\\frac{2x-1}{\\sqrt{3}}\\right)" }, { "type": "step", "result": "=-\\frac{1}{2}\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|+\\sqrt{3}\\arctan\\left(\\frac{1}{\\sqrt{3}}\\left(2x-1\\right)\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{1}{2}\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|+\\sqrt{3}\\arctan\\left(\\frac{1}{\\sqrt{3}}\\left(2x-1\\right)\\right)+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=Substitution", "practiceTopic": "Integral Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\frac{1}{2}\\ln\\left|\\frac{4}{3}+\\frac{4x^{2}-4x}{3}\\right|+\\sqrt{3}\\arctan(\\frac{1}{\\sqrt{3}}(2x-1))+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }