integral of 1/(2x^2-16)

{ "query": { "display": "$$\\int\\:\\frac{1}{2x^{2}-16}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{2x^{2}-16}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{1}{8\\sqrt{2}}(\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|)+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1}{2x^{2}-16}dx=-\\frac{1}{8\\sqrt{2}}\\left(\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|\\right)+C$$", "input": "\\int\\:\\frac{1}{2x^{2}-16}dx", "steps": [ { "type": "interim", "title": "Apply Integral Substitution", "input": "\\int\\:\\frac{1}{2x^{2}-16}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: $$x=\\frac{4}{\\sqrt{2}}u$$" ] }, { "type": "step", "primary": "For $$bx^2\\pm\\:a\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}u$$<br/>$$a=16,\\:b=2,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=\\frac{4}{\\sqrt{2}}\\quad\\Rightarrow\\quad$$substitute $$x=\\frac{4}{\\sqrt{2}}u$$" }, { "type": "interim", "title": "$$\\frac{dx}{du}=2\\sqrt{2}$$", "input": "\\frac{d}{du}\\left(\\frac{4}{\\sqrt{2}}u\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{4}{\\sqrt{2}}u:{\\quad}2^{\\frac{3}{2}}u$$", "input": "\\frac{4}{\\sqrt{2}}u", "steps": [ { "type": "interim", "title": "$$\\frac{4}{\\sqrt{2}}=2^{\\frac{3}{2}}$$", "input": "\\frac{4}{\\sqrt{2}}", "steps": [ { "type": "interim", "title": "Factor $$4:{\\quad}2^{2}$$", "steps": [ { "type": "step", "primary": "Factor $$4=2^{2}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{2^{2}}{\\sqrt{2}}" }, { "type": "interim", "title": "Cancel $$\\frac{2^{2}}{\\sqrt{2}}:{\\quad}2^{\\frac{3}{2}}$$", "input": "\\frac{2^{2}}{\\sqrt{2}}", "result": "=2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{2}=2^{\\frac{1}{2}}$$" ], "result": "=\\frac{2^{2}}{2^{\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=x^{a-b}$$", "secondary": [ "$$\\frac{2^{2}}{2^{\\frac{1}{2}}}=2^{2-\\frac{1}{2}}$$" ], "result": "=2^{2-\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$2-\\frac{1}{2}=\\frac{3}{2}$$", "result": "=2^{\\frac{3}{2}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYleDbvYvzmT313VzR+sUoldFdiKL0bo+0lLRUzNXm1IE+lnAOG+Ia4J9bL04dQr9Y0E+j94V4Px1JdcRqh4WYntFKk3fejFkyiOiq9iG9IkA+ofNvapduJ1W4ydhV/GvzDpnf1981wUSPu0q2TKJVQUySfO+PKAnmEYugQJMxbRD" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76jNun9q0ydaWyOlv11RTZCQocUa/Khw1SLN1OZoDI7yrju+5Z51e/ZZSD3gRHwjBGbX1zGEyzG4oCJZbP1Tzy2rONYbR0Uz9NaePN/Q+eVD3AEZANj/O0zvJwuhuZ1sBnAEDjbEmUeDUCgr8jdwglnamL2RaaV7qDIw2ZTB13wo=" } }, { "type": "step", "result": "=2^{\\frac{3}{2}}u" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{d}{du}\\left(2^{\\frac{3}{2}}u\\right)" }, { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2^{\\frac{3}{2}}\\frac{du}{du}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{du}{du}=1$$", "result": "=2^{\\frac{3}{2}}\\cdot\\:1" }, { "type": "interim", "title": "Simplify $$2^{\\frac{3}{2}}\\cdot\\:1:{\\quad}2\\sqrt{2}$$", "input": "2^{\\frac{3}{2}}\\cdot\\:1", "result": "=2\\sqrt{2}", "steps": [ { "type": "interim", "title": "$$2^{\\frac{3}{2}}=2\\sqrt{2}$$", "input": "2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "$$2^{\\frac{3}{2}}=2^{1+\\frac{1}{2}}$$", "result": "=2^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=2^{1}\\cdot\\:2^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74BhHVsVlToJ4si5d9eFygkV2IovRuj7SUtFTM1ebUgSzsHBJV0oRhKqf7h8oBCga/ZvNNiQXuXM/3Q/1Wd7+UvaH6vwmV87QT7wOShuZupb/hpBx7iGqK34DCOrFaZs46d4uWuWzuSbMiL3VmELCSg==" } }, { "type": "step", "result": "=2\\cdot\\:1\\cdot\\:\\sqrt{2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2\\sqrt{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74BhHVsVlToJ4si5d9eFygqJssHTganKfQXR/peqhSALNGoPE9TME3q+OPmgkv2RQaAo0m78B6eDqPnyoVzCnhP8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUlAWndrgh4akoS5OSgsbmVtdNSC7pmJktAnjSeR5Ohb9Y=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=2\\sqrt{2}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{2\\left(\\frac{4}{\\sqrt{2}}u\\right)^{2}-16}\\cdot\\:2\\sqrt{2}du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2\\left(\\frac{4}{\\sqrt{2}}u\\right)^{2}-16}\\cdot\\:2\\sqrt{2}:{\\quad}\\frac{1}{4\\sqrt{2}\\left(u^{2}-1\\right)}$$", "input": "\\frac{1}{2\\left(\\frac{4}{\\sqrt{2}}u\\right)^{2}-16}\\cdot\\:2\\sqrt{2}", "steps": [ { "type": "interim", "title": "$$\\frac{1}{2\\left(\\frac{4}{\\sqrt{2}}u\\right)^{2}-16}=\\frac{1}{16u^{2}-16}$$", "input": "\\frac{1}{2\\left(\\frac{4}{\\sqrt{2}}u\\right)^{2}-16}", "steps": [ { "type": "interim", "title": "$$2\\left(\\frac{4}{\\sqrt{2}}u\\right)^{2}=16u^{2}$$", "input": "2\\left(\\frac{4}{\\sqrt{2}}u\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{4}{\\sqrt{2}}u\\right)^{2}=2^{3}u^{2}$$", "input": "\\left(\\frac{4}{\\sqrt{2}}u\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\frac{4}{\\sqrt{2}}=2\\sqrt{2}$$", "input": "\\frac{4}{\\sqrt{2}}", "steps": [ { "type": "interim", "title": "Factor $$4:{\\quad}2^{2}$$", "steps": [ { "type": "step", "primary": "Factor $$4=2^{2}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{2^{2}}{\\sqrt{2}}" }, { "type": "interim", "title": "Cancel $$\\frac{2^{2}}{\\sqrt{2}}:{\\quad}2^{\\frac{3}{2}}$$", "input": "\\frac{2^{2}}{\\sqrt{2}}", "result": "=2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{2}=2^{\\frac{1}{2}}$$" ], "result": "=\\frac{2^{2}}{2^{\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=x^{a-b}$$", "secondary": [ "$$\\frac{2^{2}}{2^{\\frac{1}{2}}}=2^{2-\\frac{1}{2}}$$" ], "result": "=2^{2-\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$2-\\frac{1}{2}=\\frac{3}{2}$$", "result": "=2^{\\frac{3}{2}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYleDbvYvzmT313VzR+sUoldFdiKL0bo+0lLRUzNXm1IE+lnAOG+Ia4J9bL04dQr9Y0E+j94V4Px1JdcRqh4WYntFKk3fejFkyiOiq9iG9IkA+ofNvapduJ1W4ydhV/GvzDpnf1981wUSPu0q2TKJVQUySfO+PKAnmEYugQJMxbRD" } }, { "type": "interim", "title": "$$2^{\\frac{3}{2}}=2\\sqrt{2}$$", "input": "2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "$$2^{\\frac{3}{2}}=2^{1+\\frac{1}{2}}$$", "result": "=2^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=2^{1}\\cdot\\:2^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{2}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=2\\sqrt{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76jNun9q0ydaWyOlv11RTZCQocUa/Khw1SLN1OZoDI7yrju+5Z51e/ZZSD3gRHwjBBHXDUtAFqt57frcjCh0SMkeCBKuYKgaNJ253gLI69U55xDkktHQWxqGV2GFOeNWcvgrZ+z6e+oLXQViLHv62Mw==" } }, { "type": "step", "result": "=\\left(2\\sqrt{2}u\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=2^{2}\\left(\\sqrt{2}\\right)^{2}u^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\left(\\sqrt{2}\\right)^{2}:{\\quad}2$$", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(2^{\\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": "=2^{\\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": "=2", "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": "=2^{2}\\cdot\\:2u^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$2^{2}\\cdot\\:2=\\:2^{2+1}$$" ], "result": "=2^{2+1}u^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=2^{3}u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lW0oVq7Ti5Osc4m7iOY1zYaUYVxjOFqQd7yhIDM51BnehkKrn0era9rz8TlL+x/v1GrrRWlWQdoAXlGeQl5poYdEeJB8NSwK2cnf2Bc1WE1RgxASuQz3IAmMt71S18RkrPC5AnnkYlV4/CcIGzHwnD0SMibvuDzr1j4lI7sEOLk=" } }, { "type": "step", "result": "=2^{3}\\cdot\\:2u^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$2\\cdot\\:2^{3}=\\:2^{1+3}$$" ], "result": "=2^{1+3}u^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=16u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Uk+kQ8CBGnwRuIesv5nE/Ndsj3hVllRoqhnbZ8tOEdlV00rpv8+ZC6TM10tVCSHsm0ZSxSKF3As979faqmgq2hHO0oTnnZveyzJ4AtC1ZGME5ZYguuuUIH557lUx0V3PHltW3jZaoiSEyQPEPWeOuhnqUGfNyZ2pwMevAkyMU0Q=" } }, { "type": "step", "result": "=\\frac{1}{16u^{2}-16}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jg7HxbgR8hwMLRrPxj3eHwOtQHVNj8eqEZK6eV4fsGFLZS+EPy0br+W2Xn5r3ohICf2WQN9mSJxQaQ7cQX4il2BgpogRffZXrJl2fYiHyePpN/lTwGEdjn0wxfAsz3Mo3oe/oyhMy2+1TQhDBd2f9HOKYJR5Jjw/E0bW/7kWzsJaOkjHq0NvwqZQ984M9rcXvGGI0vW/vV32MwdW2mvOoPlnZoZbUE92WLARb5QYp4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=2\\sqrt{2}\\frac{1}{16u^{2}-16}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2\\sqrt{2}}{16u^{2}-16}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2\\sqrt{2}}{16u^{2}-16}" }, { "type": "interim", "title": "Factor $$16u^{2}-16:{\\quad}16\\left(u^{2}-1\\right)$$", "input": "16u^{2}-16", "result": "=\\frac{2\\sqrt{2}}{16\\left(u^{2}-1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=16u^{2}-16\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$16$$", "result": "=16\\left(u^{2}-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{\\sqrt{2}}{8\\left(u^{2}-1\\right)}" }, { "type": "interim", "title": "Factor $$8:{\\quad}2^{3}$$", "steps": [ { "type": "step", "primary": "Factor $$8=2^{3}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{2}}{2^{3}\\left(u^{2}-1\\right)}" }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{2}}{2^{3}\\left(u^{2}-1\\right)}:{\\quad}\\frac{1}{2^{\\frac{5}{2}}\\left(u^{2}-1\\right)}$$", "input": "\\frac{\\sqrt{2}}{2^{3}\\left(u^{2}-1\\right)}", "result": "=\\frac{1}{2^{\\frac{5}{2}}\\left(u^{2}-1\\right)}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{2}=2^{\\frac{1}{2}}$$" ], "result": "=\\frac{2^{\\frac{1}{2}}}{2^{3}\\left(u^{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": [ "$$\\frac{2^{\\frac{1}{2}}}{2^{3}}=\\frac{1}{2^{3-\\frac{1}{2}}}$$" ], "result": "=\\frac{1}{2^{-\\frac{1}{2}+3}\\left(u^{2}-1\\right)}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$3-\\frac{1}{2}=\\frac{5}{2}$$", "result": "=\\frac{1}{2^{\\frac{5}{2}}\\left(u^{2}-1\\right)}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYv1j4TgIZ9XijfbXZdkXhB94YNb1okM2P6yieeBIkGjS3XeO2tIUPH5Q2xrCOU6NXUkdpGrFPKTOn6vuS9dML0iSY6+96dInyvdoOptlAniXEF1LGYWni+ewe7ssB3Cq0PC30sSftAIFS6Qkpy19IkrfcvtK1wS5X9djgJbBX+NYQuod4MFxa/IqIEw7tswnJjVg0XyEA1UWimb6zrLL+1g=" } }, { "type": "interim", "title": "$$2^{\\frac{5}{2}}=2^{2}\\sqrt{2}$$", "input": "2^{\\frac{5}{2}}", "steps": [ { "type": "step", "primary": "$$2^{\\frac{5}{2}}=2^{2+\\frac{1}{2}}$$", "result": "=2^{2+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=2^{2}\\cdot\\:2^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=2^{2}\\sqrt{2}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{1}{2^{2}\\sqrt{2}\\left(u^{2}-1\\right)}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\frac{1}{4\\sqrt{2}\\left(u^{2}-1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{4\\sqrt{2}\\left(u^{2}-1\\right)}du" } ], "meta": { "interimType": "Integral Substitution 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{4\\sqrt{2}\\left(u^{2}-1\\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": "=\\frac{1}{4\\sqrt{2}}\\cdot\\:\\int\\:\\frac{1}{u^{2}-1}du" }, { "type": "interim", "title": "Factor $$u^{2}-1:{\\quad}-\\left(-u^{2}+1\\right)$$", "input": "u^{2}-1", "steps": [ { "type": "step", "primary": "Rewrite $$u^{2}$$ as $$-\\left(-u^{2}\\right)$$", "result": "=-\\left(-u^{2}\\right)-1" }, { "type": "step", "primary": "Factor out $$-1$$", "result": "=-\\left(-u^{2}+1\\right)" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "=\\frac{1}{4\\sqrt{2}}\\cdot\\:\\int\\:\\frac{1}{-\\left(-u^{2}+1\\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": "=\\frac{1}{4\\sqrt{2}}\\left(-\\int\\:\\frac{1}{-u^{2}+1}du\\right)" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{-u^{2}+1}du=\\frac{\\ln\\left|u+1\\right|}{2}-\\frac{\\ln\\left|u-1\\right|}{2}$$", "result": "=\\frac{1}{4\\sqrt{2}}\\left(-\\left(\\frac{\\ln\\left|u+1\\right|}{2}-\\frac{\\ln\\left|u-1\\right|}{2}\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=\\frac{\\sqrt{2}}{4}x$$", "result": "=\\frac{1}{4\\sqrt{2}}\\left(-\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{1}{4\\sqrt{2}}\\left(-\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)\\right):{\\quad}-\\frac{1}{8\\sqrt{2}}\\left(\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|\\right)$$", "input": "\\frac{1}{4\\sqrt{2}}\\left(-\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)\\right)", "result": "=-\\frac{1}{8\\sqrt{2}}\\left(\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{4\\sqrt{2}}\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)}{4\\sqrt{2}}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)=\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}$$", "input": "1\\cdot\\:\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)=\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)$$", "result": "=\\left(\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "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" } }, { "type": "step", "result": "=-\\frac{\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}}{4\\sqrt{2}}" }, { "type": "interim", "title": "$$\\frac{\\sqrt{2}}{4}x=\\frac{x}{2\\sqrt{2}}$$", "input": "\\frac{\\sqrt{2}}{4}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{2}x}{4}" }, { "type": "interim", "title": "Factor $$4:{\\quad}2^{2}$$", "steps": [ { "type": "step", "primary": "Factor $$4=2^{2}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{2}x}{2^{2}}" }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{2}x}{2^{2}}:{\\quad}\\frac{x}{2^{\\frac{3}{2}}}$$", "input": "\\frac{\\sqrt{2}x}{2^{2}}", "result": "=\\frac{x}{2^{\\frac{3}{2}}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{2}=2^{\\frac{1}{2}}$$" ], "result": "=\\frac{2^{\\frac{1}{2}}x}{2^{2}}", "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": [ "$$\\frac{2^{\\frac{1}{2}}}{2^{2}}=\\frac{1}{2^{2-\\frac{1}{2}}}$$" ], "result": "=\\frac{x}{2^{2-\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$2-\\frac{1}{2}=\\frac{3}{2}$$", "result": "=\\frac{x}{2^{\\frac{3}{2}}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgwQCZLO5YpdyMgGXb5nuntU4kXWq0uVmlMI1nL7h2Z4zvYaGmwZlvy5qMc82VI6dZ3i+5OpU9m5VLb7yq2yta+EJcrPfb+hjAR04IP0d5l21sD7NfhsPe7eDHrmjY0mE2J/JAWPJQkixCwxTVklZhMMEAmSzuWKXcjIBl2+Z7p7aDkLa+SNXTXb5AQTFxkRZg==" } }, { "type": "interim", "title": "$$2^{\\frac{3}{2}}=2\\sqrt{2}$$", "input": "2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "$$2^{\\frac{3}{2}}=2^{1+\\frac{1}{2}}$$", "result": "=2^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=2^{1}\\cdot\\:2^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{2}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{x}{2\\sqrt{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMcf/wFgWNNYOxYZzrcQWEhMJQJZuTAY5js+oqjdT8ksliKOSAqq46v09ENAWjbyT+nbKyzRGn37jQ8thU9HS1iRTW26qciuyUBGXQExCUedYHdsJW1Io4s1nV/QEGcdaxpGghgGYNoF4STXU8l/ICIBYlSN7rzdW8vgDfTX46ChV" } }, { "type": "step", "result": "=-\\frac{\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|}{2}-\\frac{\\ln\\left|\\frac{\\sqrt{2}}{4}x-1\\right|}{2}}{4\\sqrt{2}}" }, { "type": "interim", "title": "$$\\frac{\\sqrt{2}}{4}x=\\frac{x}{2\\sqrt{2}}$$", "input": "\\frac{\\sqrt{2}}{4}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{2}x}{4}" }, { "type": "interim", "title": "Factor $$4:{\\quad}2^{2}$$", "steps": [ { "type": "step", "primary": "Factor $$4=2^{2}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{2}x}{2^{2}}" }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{2}x}{2^{2}}:{\\quad}\\frac{x}{2^{\\frac{3}{2}}}$$", "input": "\\frac{\\sqrt{2}x}{2^{2}}", "result": "=\\frac{x}{2^{\\frac{3}{2}}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{2}=2^{\\frac{1}{2}}$$" ], "result": "=\\frac{2^{\\frac{1}{2}}x}{2^{2}}", "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": [ "$$\\frac{2^{\\frac{1}{2}}}{2^{2}}=\\frac{1}{2^{2-\\frac{1}{2}}}$$" ], "result": "=\\frac{x}{2^{2-\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$2-\\frac{1}{2}=\\frac{3}{2}$$", "result": "=\\frac{x}{2^{\\frac{3}{2}}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgwQCZLO5YpdyMgGXb5nuntU4kXWq0uVmlMI1nL7h2Z4zvYaGmwZlvy5qMc82VI6dZ3i+5OpU9m5VLb7yq2yta+EJcrPfb+hjAR04IP0d5l21sD7NfhsPe7eDHrmjY0mE2J/JAWPJQkixCwxTVklZhMMEAmSzuWKXcjIBl2+Z7p7aDkLa+SNXTXb5AQTFxkRZg==" } }, { "type": "interim", "title": "$$2^{\\frac{3}{2}}=2\\sqrt{2}$$", "input": "2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "$$2^{\\frac{3}{2}}=2^{1+\\frac{1}{2}}$$", "result": "=2^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=2^{1}\\cdot\\:2^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{2}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{x}{2\\sqrt{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMcf/wFgWNNYOxYZzrcQWEhMJQJZuTAY5js+oqjdT8ksliKOSAqq46v09ENAWjbyT+nbKyzRGn37jQ8thU9HS1iRTW26qciuyUBGXQExCUedYHdsJW1Io4s1nV/QEGcdaxpGghgGYNoF4STXU8l/ICIBYlSN7rzdW8vgDfTX46ChV" } }, { "type": "step", "result": "=-\\frac{\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|}{2}-\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{2}}{4\\sqrt{2}}" }, { "type": "interim", "title": "Join $$\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|}{2}-\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{2}:{\\quad}\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{2}$$", "input": "\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|}{2}-\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{2}", "result": "=-\\frac{\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{2}}{4\\sqrt{2}}", "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{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{2}}{4\\sqrt{2}}:{\\quad}\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{8\\sqrt{2}}$$", "input": "\\frac{\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{2}}{4\\sqrt{2}}", "result": "=-\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{8\\sqrt{2}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{2\\cdot\\:4\\sqrt{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|}{8\\sqrt{2}}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{1}{8\\sqrt{2}}\\left(\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-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}{8\\sqrt{2}}\\left(\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-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", "practiceTopic": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=-\\frac{1}{8\\sqrt{2}}(\\ln\\left|\\frac{x}{2\\sqrt{2}}+1\\right|-\\ln\\left|\\frac{x}{2\\sqrt{2}}-1\\right|)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }