integral of 1/(sqrt(x^2+36))

{ "query": { "display": "$$\\int\\:\\frac{1}{\\sqrt{x^{2}+36}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{\\sqrt{x^{2}+36}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\ln(\\frac{1}{6}\\left|x+\\sqrt{36+x^{2}}\\right|)+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1}{\\sqrt{x^{2}+36}}dx=\\ln\\left(\\frac{1}{6}\\left|x+\\sqrt{36+x^{2}}\\right|\\right)+C$$", "input": "\\int\\:\\frac{1}{\\sqrt{x^{2}+36}}dx", "steps": [ { "type": "interim", "title": "Apply Trigonometric Substitution", "input": "\\int\\:\\frac{1}{\\sqrt{x^{2}+36}}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$" }, { "type": "step", "primary": "For $$\\sqrt{bx^2+a}\\:$$substitute $$x=\\frac{\\sqrt{a}}{\\sqrt{b}}\\tan\\left(u\\right)$$<br/>$$a=36,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=6\\quad\\Rightarrow\\quad$$substitute $$x=6\\tan\\left(u\\right)$$" }, { "type": "interim", "title": "$$\\frac{dx}{du}=6\\sec^{2}\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(6\\tan\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{du}\\left(\\tan\\left(u\\right)\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\tan\\left(u\\right)\\right)=\\sec^{2}\\left(u\\right)$$", "result": "=6\\sec^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYomNSCISYRDoYfA0O6PJkrOQp7tdIFyr1eVqMMLZHDTG7JiujyaxykLR7Tv0Nr+wOL61PtlKAyohFUmvnBv69gsHk1eLttM1tdVeTyzsC+lrgYQxnkXodDkwXaY3EUnksbitWfUmjL+8riWlL70A2fk=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=6\\sec^{2}\\left(u\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{\\left(6\\tan\\left(u\\right)\\right)^{2}+36}}\\cdot\\:6\\sec^{2}\\left(u\\right)du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sqrt{\\left(6\\tan\\left(u\\right)\\right)^{2}+36}}\\cdot\\:6\\sec^{2}\\left(u\\right):{\\quad}\\sec\\left(u\\right)$$", "input": "\\frac{1}{\\sqrt{\\left(6\\tan\\left(u\\right)\\right)^{2}+36}}\\cdot\\:6\\sec^{2}\\left(u\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\sqrt{\\left(6\\tan\\left(u\\right)\\right)^{2}+36}}=\\frac{1}{6\\sqrt{\\tan^{2}\\left(u\\right)+1}}$$", "input": "\\frac{1}{\\sqrt{\\left(6\\tan\\left(u\\right)\\right)^{2}+36}}", "steps": [ { "type": "interim", "title": "$$\\sqrt{\\left(6\\tan\\left(u\\right)\\right)^{2}+36}=6\\sqrt{\\tan^{2}\\left(u\\right)+1}$$", "input": "\\sqrt{\\left(6\\tan\\left(u\\right)\\right)^{2}+36}", "steps": [ { "type": "interim", "title": "$$\\left(6\\tan\\left(u\\right)\\right)^{2}=36\\tan^{2}\\left(u\\right)$$", "input": "\\left(6\\tan\\left(u\\right)\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=6^{2}\\tan^{2}\\left(u\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$6^{2}=36$$", "result": "=36\\tan^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7frcS85rO7SkTyOrLeTXNyXyRHuGw7+tM5METTDj6vVFh1WQA7gcn24yur8P2LLUIlPJCKAYbTy+6j2BfEpwMcJtAiLfoOGOT+Nk4bif2GTQ5db1Kv15xa/ljllzM0/zqXM61ctLOA6XhGZhfRVLmqQ==" } }, { "type": "step", "result": "=\\sqrt{36\\tan^{2}\\left(u\\right)+36}" }, { "type": "interim", "title": "Factor $$36\\tan^{2}\\left(u\\right)+36:{\\quad}36\\left(\\tan^{2}\\left(u\\right)+1\\right)$$", "input": "36\\tan^{2}\\left(u\\right)+36", "result": "=\\sqrt{36\\left(\\tan^{2}\\left(u\\right)+1\\right)}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=36\\tan^{2}\\left(u\\right)+36\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$36$$", "result": "=36\\left(\\tan^{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{36}\\sqrt{\\tan^{2}\\left(u\\right)+1}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{36}=6$$", "input": "\\sqrt{36}", "result": "=6\\sqrt{\\tan^{2}\\left(u\\right)+1}", "steps": [ { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7piv/c6VAEdl4mVsOUXPeIwQJ5IXDKzcIEtGXizOxWgAgJ/ZZA32ZInFBpDtxBfiK/oo8C/X8WsWQEdYD96Bd14Wcgy7Rm9AE94dPz/sXw50/y9DKGIPglJ+qMi9xDu2KYiIZ39fDafi8W0WPzEdoE2Rocexkc4a9nSlmi6EWcEIcuYdftGYMmy+Hm6T7Axc5bGUhD9+gDQg079F600Xu5Q==" } }, { "type": "step", "result": "=\\frac{1}{6\\sqrt{\\tan^{2}\\left(u\\right)+1}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGWw6+2nDVMHc1Edy4v+YrEBwRRbUxdy2KFEvjH3nPo9uCUCWbkwGOY7PqKo3U/JLJWBHQhFdaAqRxGkKPanKlN4GWmGbPD50cAE2BtjuQYyCX80FwFd5G4mdz723GaCnMt//gHf/VmzVnT/82n2sEPdiIhnf18Np+LxbRY/MR2gTvkb2U5tPE4cxauvDnn1F9yWB36ZeIFkHQITTNY8khkh72zpVCRYQQH/KPvv2unVMJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=6\\cdot\\:\\frac{1}{6\\sqrt{\\tan^{2}\\left(u\\right)+1}}\\sec^{2}\\left(u\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:6\\sec^{2}\\left(u\\right)}{6\\sqrt{\\tan^{2}\\left(u\\right)+1}}" }, { "type": "step", "primary": "Cancel the common factor: $$6$$", "result": "=\\frac{1\\cdot\\:\\sec^{2}\\left(u\\right)}{\\sqrt{\\tan^{2}\\left(u\\right)+1}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\sec^{2}\\left(u\\right)=\\sec^{2}\\left(u\\right)$$", "result": "=\\frac{\\sec^{2}\\left(u\\right)}{\\sqrt{\\tan^{2}\\left(u\\right)+1}}" }, { "type": "interim", "title": "Simplify $$\\sqrt{\\tan^{2}\\left(u\\right)+1}:{\\quad}\\sqrt{\\sec^{2}\\left(u\\right)}$$", "input": "\\sqrt{\\tan^{2}\\left(u\\right)+1}", "result": "=\\frac{\\sec^{2}\\left(u\\right)}{\\sqrt{\\sec^{2}\\left(u\\right)}}", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\tan^{2}\\left(x\\right)+1=\\sec^{2}\\left(x\\right)$$", "result": "=\\sqrt{\\sec^{2}\\left(u\\right)}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "$$\\sqrt{\\sec^{2}\\left(u\\right)}=\\sec\\left(u\\right)$$", "input": "\\sqrt{\\sec^{2}\\left(u\\right)}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "result": "=\\sec\\left(u\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pNgPPjxfgQQy4WKXvL0D2pdjjcAkasS081pL8yxvuR+rju+5Z51e/ZZSD3gRHwjBjzwMNNH1RyBVkalYf5Lb0rtCR5dIjxQ5ASg+ZPFVSseQCzXlmwP37ag6r9r431jfTfl1bWwdKMwj8GbMLAzUXA==" } }, { "type": "step", "result": "=\\frac{\\sec^{2}\\left(u\\right)}{\\sec\\left(u\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sec\\left(u\\right)$$", "result": "=\\sec\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\sec\\left(u\\right)du" } ], "meta": { "interimType": "Integral Trig Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73YBJdU3jgZIBfx9KjN2Y11Ci7uZiZUj5yHewJVPa5m8SUM9pakkKILvT6Fs/PM359VEgBuKxZgOTX2ljN4yVJ9qQggBPzB4Qayiyi1+p6hLgSEHRda+G5BHM5FRE2/NG59vNZ/YD5VvkYh2E6N1WmVFKk3fejFkyiOiq9iG9IkAsORR8wbmFxGvaMhU0Myw3jlcNomyisbrFmsXVa8cmNwohx3JNTaptNuNW2szzOqF" } }, { "type": "step", "result": "=\\int\\:\\sec\\left(u\\right)du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\sec\\left(u\\right)du=\\ln\\left|\\tan\\left(u\\right)+\\sec\\left(u\\right)\\right|$$", "result": "=\\ln\\left|\\tan\\left(u\\right)+\\sec\\left(u\\right)\\right|" }, { "type": "step", "primary": "Substitute back $$u=\\arctan\\left(\\frac{1}{6}x\\right)$$", "result": "=\\ln\\left|\\tan\\left(\\arctan\\left(\\frac{1}{6}x\\right)\\right)+\\sec\\left(\\arctan\\left(\\frac{1}{6}x\\right)\\right)\\right|" }, { "type": "interim", "title": "Simplify $$\\ln\\left|\\tan\\left(\\arctan\\left(\\frac{1}{6}x\\right)\\right)+\\sec\\left(\\arctan\\left(\\frac{1}{6}x\\right)\\right)\\right|:{\\quad}\\ln\\left(\\frac{1}{6}\\left|x+\\sqrt{36+x^{2}}\\right|\\right)$$", "input": "\\ln\\left|\\tan\\left(\\arctan\\left(\\frac{1}{6}x\\right)\\right)+\\sec\\left(\\arctan\\left(\\frac{1}{6}x\\right)\\right)\\right|", "result": "=\\ln\\left(\\frac{1}{6}\\left|x+\\sqrt{36+x^{2}}\\right|\\right)", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\sec\\left(\\arctan\\left(x\\right)\\right)=\\sqrt{1+x^{2}}$$", "result": "=\\ln\\left|\\tan\\left(\\arctan\\left(\\frac{1}{6}x\\right)\\right)+\\sqrt{1+\\left(\\frac{1}{6}x\\right)^{2}}\\right|" }, { "type": "step", "primary": "Use the following identity: $$\\tan\\left(\\arctan\\left(x\\right)\\right)=x$$", "result": "=\\ln\\left|\\frac{1}{6}x+\\sqrt{1+\\left(\\frac{1}{6}x\\right)^{2}}\\right|" }, { "type": "interim", "title": "$$\\sqrt{1+\\left(\\frac{1}{6}x\\right)^{2}}=\\frac{\\sqrt{36+x^{2}}}{6}$$", "input": "\\sqrt{1+\\left(\\frac{1}{6}x\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{1}{6}x\\right)^{2}=\\frac{1}{36}x^{2}$$", "input": "\\left(\\frac{1}{6}x\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=\\left(\\frac{1}{6}\\right)^{2}x^{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": [ "$$\\left(\\frac{1}{6}\\right)^{2}=\\frac{1^{2}}{6^{2}}$$" ], "result": "=\\frac{1^{2}}{6^{2}}x^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{1}{6^{2}}x^{2}" }, { "type": "step", "primary": "$$6^{2}=36$$", "result": "=\\frac{1}{36}x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7U8GnrOzd32HCt6sJBdA59+iEPDD5lvIAC9CzFeUpV5JwkKGJWEPFPk38sdJMsyPIxJQc4K7spNw6Naq/rf5rWYdEeJB8NSwK2cnf2Bc1WE1YXCDedPcDfP5ynHQiWJLfM74kY3UEN+IaALM67RsB7xM8P8KKjpcuzzFPXmLNmwA=" } }, { "type": "step", "result": "=\\sqrt{1+\\frac{1}{36}x^{2}}" }, { "type": "interim", "title": "Join $$1+\\frac{1}{36}x^{2}:{\\quad}\\frac{36+x^{2}}{36}$$", "input": "1+\\frac{1}{36}x^{2}", "result": "=\\sqrt{\\frac{36+x^{2}}{36}}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{1}{36}x^{2}\\::{\\quad}\\frac{x^{2}}{36}$$", "input": "\\frac{1}{36}x^{2}", "result": "=1+\\frac{x^{2}}{36}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:x^{2}}{36}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{2}=x^{2}$$", "result": "=\\frac{x^{2}}{36}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:36}{36}$$", "result": "=\\frac{1\\cdot\\:36}{36}+\\frac{x^{2}}{36}" }, { "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\\:36+x^{2}}{36}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:36=36$$", "result": "=\\frac{36+x^{2}}{36}" } ], "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{36+x^{2}}}{\\sqrt{36}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "interim", "title": "$$\\sqrt{36}=6$$", "input": "\\sqrt{36}", "result": "=\\frac{\\sqrt{36+x^{2}}}{6}", "steps": [ { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "N/A" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7724VmTHcWiYrG8awSZbuDBiiHpkM4C1WF4aWPzkHiqwtOtZYwUjyXhDTsNnn6ElrSGpx9e3Lolh3WlYiBDdmOMnY637hF/ruvT6LyHBg36241YxxBC+sxUOAsLBE1I/+k4diZxjrcGW7LSEnYl1L0Irim2ceBtaoJMikFt22dA+97gdKeL0xurKNTRc16wz7N6TPF3jrMjxd67ZE1PirSrCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\ln\\left|\\frac{1}{6}x+\\frac{\\sqrt{36+x^{2}}}{6}\\right|" }, { "type": "interim", "title": "Join $$\\frac{1}{6}x+\\frac{\\sqrt{36+x^{2}}}{6}:{\\quad}\\frac{x+\\sqrt{x^{2}+36}}{6}$$", "input": "\\frac{1}{6}x+\\frac{\\sqrt{36+x^{2}}}{6}", "result": "=\\ln\\left|\\frac{x+\\sqrt{36+x^{2}}}{6}\\right|", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{1}{6}x\\::{\\quad}\\frac{x}{6}$$", "input": "\\frac{1}{6}x", "result": "=\\frac{x}{6}+\\frac{\\sqrt{36+x^{2}}}{6}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:x}{6}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=\\frac{x}{6}" } ], "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{x+\\sqrt{36+x^{2}}}{6}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\left|\\frac{x+\\sqrt{36+x^{2}}}{6}\\right|:{\\quad}\\frac{\\left|x+\\sqrt{x^{2}+36}\\right|}{6}$$", "input": "\\left|\\frac{x+\\sqrt{36+x^{2}}}{6}\\right|", "result": "=\\ln\\left(\\frac{\\left|x+\\sqrt{36+x^{2}}\\right|}{6}\\right)", "steps": [ { "type": "step", "primary": "Apply absolute rule: $$\\left|\\frac{a}{b}\\right|\\:=\\frac{\\left|a\\right|}{\\left|b\\right|}$$", "secondary": [ "$$\\left|\\frac{x+\\sqrt{36+x^{2}}}{6}\\right|=\\frac{\\left|x+\\sqrt{x^{2}+36}\\right|}{\\left|6\\right|}$$" ], "result": "=\\frac{\\left|x+\\sqrt{36+x^{2}}\\right|}{\\left|6\\right|}" }, { "type": "step", "primary": "Apply absolute rule: $$\\left|a\\right|=a,\\:a\\ge0$$", "secondary": [ "$$\\left|6\\right|=6$$" ], "result": "=\\frac{\\left|x+\\sqrt{36+x^{2}}\\right|}{6}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\ln\\left(\\frac{1}{6}\\left|x+\\sqrt{36+x^{2}}\\right|\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74fjd84dnSZau8q2jLZK3XJGaboEorFFctF1ewhMLXxbMWuYsakcThPmX76KDoAM8a69+pWOQyx1BwTzwilUoIjTxXCoT0R/r/J4nJ17JL5IEWuLCXstQFFOxghO3RS4O/aL2Coo0GISQwm8bv5wDiXZgQCrkttosGhjXZUQCTH4pNvErlLU/iHyqTC39Ne5LL0oIwxIf/UmqEITFkkYXbKExQUMU4zW0Le6rXPRlWkceKYFFhGodJZ4mS5Q+F9ML4o6goNu9Z9w5212GyAhFfqITFjjyitWeIizVYbJERrLJ9xQVN6wISd6cZA3EwtPF97dael8Q1nwMfN0s0HL3JCEpoQIKaTwcekRCOvAbkoDm8RQrOC09tvNLWdNDDl9E" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\ln\\left(\\frac{1}{6}\\left|x+\\sqrt{36+x^{2}}\\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=Trig%20Power%20Multiplication", "practiceTopic": "Integral Trig Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\ln(\\frac{1}{6}\\left|x+\\sqrt{36+x^{2}}\\right|)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }