integral of 1/(sqrt(x^2+3))

{ "query": { "display": "$$\\int\\:\\frac{1}{\\sqrt{x^{2}+3}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{\\sqrt{x^{2}+3}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\ln\\left|\\frac{1}{\\sqrt{3}}x+\\sqrt{\\frac{1}{3}(3+x^{2})}\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1}{\\sqrt{x^{2}+3}}dx=\\ln\\left|\\frac{1}{\\sqrt{3}}x+\\sqrt{\\frac{1}{3}\\left(3+x^{2}\\right)}\\right|+C$$", "input": "\\int\\:\\frac{1}{\\sqrt{x^{2}+3}}dx", "steps": [ { "type": "interim", "title": "Apply Trigonometric Substitution", "input": "\\int\\:\\frac{1}{\\sqrt{x^{2}+3}}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/>$$a=3,\\:b=1,\\:\\frac{\\sqrt{a}}{\\sqrt{b}}=\\sqrt{3}\\quad\\Rightarrow\\quad$$substitute $$x=\\sqrt{3}\\tan\\left(u\\right)$$" }, { "type": "interim", "title": "$$\\frac{dx}{du}=\\sqrt{3}\\sec^{2}\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=\\sqrt{3}\\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": "=\\sqrt{3}\\sec^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiRWf3eWIAF4vZjHMhbuLW9Cc/C1t1xZcTo8/4x8UlP0SfFZOsLyo1Z07cFk9fG0J4Tpirx94ZbaetCgA/t6c3y+tT7ZSgMqIRVJr5wb+vYLB5NXi7bTNbXVXk8s7Avpa3L8n7qMmnFGYGAhcUS0oPtVm2iAQ4qu9Wm8icUYLcO5sIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\sqrt{3}\\sec^{2}\\left(u\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)^{2}+3}}\\sqrt{3}\\sec^{2}\\left(u\\right)du" }, { "type": "interim", "title": "Simplify $$\\frac{1}{\\sqrt{\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)^{2}+3}}\\sqrt{3}\\sec^{2}\\left(u\\right):{\\quad}\\sec\\left(u\\right)$$", "input": "\\frac{1}{\\sqrt{\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)^{2}+3}}\\sqrt{3}\\sec^{2}\\left(u\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\sqrt{\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)^{2}+3}}=\\frac{1}{\\sqrt{3\\tan^{2}\\left(u\\right)+3}}$$", "input": "\\frac{1}{\\sqrt{\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)^{2}+3}}", "steps": [ { "type": "interim", "title": "$$\\sqrt{\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)^{2}+3}=\\sqrt{3\\tan^{2}\\left(u\\right)+3}$$", "input": "\\sqrt{\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)^{2}+3}", "steps": [ { "type": "interim", "title": "$$\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)^{2}=3\\tan^{2}\\left(u\\right)$$", "input": "\\left(\\sqrt{3}\\tan\\left(u\\right)\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=\\left(\\sqrt{3}\\right)^{2}\\tan^{2}\\left(u\\right)", "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": "=3\\tan^{2}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7j3/VvRzmILxr9D21nbmDVd+qeRw8qwJVMcDIlqYjw7oDnzlbPZjyKgy1eUCFsLd5+syYNn42y6JC0fp/yji1tKlm8S/dpnAYIrx0z4b0scz9+/Q6Bo7D02RficGzIS3XDeU3ZynZRCJ3yY+or/SDRoiobvoYuxdIB1LMc+5lVss=" } }, { "type": "step", "result": "=\\sqrt{3\\tan^{2}\\left(u\\right)+3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WXStXFVx75hYDCxA7hRbNMJS/oQQrZDdJ1e8Z85/UbZBWBlrb/kgCHcV+e0/Hw6Qo5FYteSPKwXny4uCMrdsK46NcdjPdkfroAiG+IKxFFUuWBpNMyLvOqKx16Epj7Tvz0u0oYyhHJnT30W0t41VHHV/8pSoZFRk9+GOhg6vo2z6+YF3Ll6i/cu0enGduMDbwq/hcze4n+guH/sOjgGJf7CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{1}{\\sqrt{3\\tan^{2}\\left(u\\right)+3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGVcdKQwCpqqI8BNRp4o/6blQiy0evvC+tgIgDNa55GBuVdNK6b/PmQukzNdLVQkh7NMQ0vmOWo9IZfhei7w2gJbGsKuJngDFSP2zZgEAcUxX4ugAu8oC3lqnejxsnQG+VkeCBKuYKgaNJ253gLI69U4q/xpezAluVXZUyfkUjklzdX/ylKhkVGT34Y6GDq+jbImwWJ0aNnS8SeJoFU4iX2llxEurNkDHnVInzYEsj+NDnzL4rjX4A6T5L5+ZNMTrIQ==" } }, { "type": "step", "result": "=\\sqrt{3}\\frac{1}{\\sqrt{3\\tan^{2}\\left(u\\right)+3}}\\sec^{2}\\left(u\\right)" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\sqrt{3}\\sec^{2}\\left(u\\right)}{\\sqrt{3\\tan^{2}\\left(u\\right)+3}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\sqrt{3}=\\sqrt{3}$$", "result": "=\\frac{\\sqrt{3}\\sec^{2}\\left(u\\right)}{\\sqrt{3\\tan^{2}\\left(u\\right)+3}}" }, { "type": "interim", "title": "$$\\sqrt{3\\tan^{2}\\left(u\\right)+3}=\\sqrt{3}\\sec\\left(u\\right)$$", "input": "\\sqrt{3\\tan^{2}\\left(u\\right)+3}", "steps": [ { "type": "interim", "title": "Simplify $$3\\tan^{2}\\left(u\\right)+3:{\\quad}3\\sec^{2}\\left(u\\right)$$", "input": "3\\tan^{2}\\left(u\\right)+3", "result": "=\\sqrt{3\\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": "=3\\sec^{2}\\left(u\\right)" }, { "type": "interim", "title": "Factor $$3\\tan^{2}\\left(u\\right)+3:{\\quad}3\\left(\\tan^{2}\\left(u\\right)+1\\right)$$", "input": "3\\tan^{2}\\left(u\\right)+3", "result": "=3\\left(\\tan^{2}\\left(u\\right)+1\\right)", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=3\\tan^{2}\\left(u\\right)+3\\cdot\\:1" }, { "type": "step", "primary": "Factor out common term $$3$$", "result": "=3\\left(\\tan^{2}\\left(u\\right)+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify 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{3}\\sqrt{\\sec^{2}\\left(u\\right)}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "secondary": [ "$$\\sqrt{\\sec^{2}\\left(u\\right)}=\\sec\\left(u\\right)$$" ], "result": "=\\sqrt{3}\\sec\\left(u\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fgZ/rg6iONxdyi2Gi/MUnvqwvdEFUiGEK+he60jlCTYDnzlbPZjyKgy1eUCFsLd5wAcG6PUp3wtYkUmqiTDOUrGvdhATgtRoATpTb/bvUWtLvgZqMXjpaUERLlONz8hZ5xYNh+SMXC+kVNsvVkahQtPMGShBDhE1O/Qtu0vCEBywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}\\sec^{2}\\left(u\\right)}{\\sqrt{3}\\sec\\left(u\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "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+4/5L7s73YBJdU3jgZIBfx9KjN2Y10GF9blvPNWLy31GS+koYj0K576WENGsoCX8pronHaqWdYx0sZocxGfTy5CECuSj+5gkc6tjzNT9fy/feSKnKbgO3d78O3qHFKyNE4EBiKgvwZIK+lGnyKAShxmbKuLFVrvbBmbuQNTF0TphKZ8RuvasX7shWA+Uqvg5pyoDL2WEqgd8xT0wj1G4DEH8QCPRFe3EQh3uVlxId1MCwC+ZNkW" } }, { "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 $$u=\\arctan\\left(\\frac{1}{\\sqrt{3}}x\\right)$$", "result": "=\\ln\\left|\\tan\\left(\\arctan\\left(\\frac{1}{\\sqrt{3}}x\\right)\\right)+\\sec\\left(\\arctan\\left(\\frac{1}{\\sqrt{3}}x\\right)\\right)\\right|" }, { "type": "interim", "title": "Simplify $$\\ln\\left|\\tan\\left(\\arctan\\left(\\frac{1}{\\sqrt{3}}x\\right)\\right)+\\sec\\left(\\arctan\\left(\\frac{1}{\\sqrt{3}}x\\right)\\right)\\right|:{\\quad}\\ln\\left|\\frac{1}{\\sqrt{3}}x+\\sqrt{\\frac{1}{3}\\left(3+x^{2}\\right)}\\right|$$", "input": "\\ln\\left|\\tan\\left(\\arctan\\left(\\frac{1}{\\sqrt{3}}x\\right)\\right)+\\sec\\left(\\arctan\\left(\\frac{1}{\\sqrt{3}}x\\right)\\right)\\right|", "result": "=\\ln\\left|\\frac{1}{\\sqrt{3}}x+\\sqrt{\\frac{1}{3}\\left(3+x^{2}\\right)}\\right|", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\tan\\left(\\arctan\\left(x\\right)\\right)=x$$", "result": "=\\ln\\left|\\frac{1}{\\sqrt{3}}x+\\sec\\left(\\arctan\\left(\\frac{1}{\\sqrt{3}}x\\right)\\right)\\right|" }, { "type": "step", "primary": "Use the following identity: $$\\sec\\left(\\arctan\\left(x\\right)\\right)=\\sqrt{1+x^{2}}$$", "result": "=\\ln\\left|\\frac{1}{\\sqrt{3}}x+\\sqrt{1+\\left(\\frac{1}{\\sqrt{3}}x\\right)^{2}}\\right|" }, { "type": "interim", "title": "$$\\sqrt{1+\\left(\\frac{1}{\\sqrt{3}}x\\right)^{2}}=\\sqrt{\\frac{3+x^{2}}{3}}$$", "input": "\\sqrt{1+\\left(\\frac{1}{\\sqrt{3}}x\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{1}{\\sqrt{3}}x\\right)^{2}=\\frac{1}{3}x^{2}$$", "input": "\\left(\\frac{1}{\\sqrt{3}}x\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "result": "=\\left(\\frac{1}{\\sqrt{3}}\\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": [ "$$\\left(\\frac{1}{\\sqrt{3}}\\right)^{2}=\\frac{1^{2}}{\\left(\\sqrt{3}\\right)^{2}}$$" ], "result": "=\\frac{1^{2}}{\\left(\\sqrt{3}\\right)^{2}}x^{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{1^{2}}{3}x^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{1}{3}x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qtWSHifc9AkvQ1rSvIucJKwwC9SkE2IzPe7p3VjFsF/ehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpk9wIXk+C4RVUbQO/kK9NgLMvNqk2VBIubwkA9Jz/J2mRHQ76YQbq9OExnn4OsamScYFvOb9TkB1l6yiLaieLaXOq6mQ2VgXloxWIHnj1yWQ" } }, { "type": "step", "result": "=\\sqrt{1+\\frac{1}{3}x^{2}}" }, { "type": "interim", "title": "Join $$1+\\frac{1}{3}x^{2}:{\\quad}\\frac{3+x^{2}}{3}$$", "input": "1+\\frac{1}{3}x^{2}", "result": "=\\sqrt{\\frac{3+x^{2}}{3}}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{1}{3}x^{2}\\::{\\quad}\\frac{x^{2}}{3}$$", "input": "\\frac{1}{3}x^{2}", "result": "=1+\\frac{x^{2}}{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:x^{2}}{3}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{2}=x^{2}$$", "result": "=\\frac{x^{2}}{3}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=\\frac{1\\cdot\\:3}{3}+\\frac{x^{2}}{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+x^{2}}{3}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3+x^{2}}{3}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7724VmTHcWiYrG8awSZbuDER0O+mEG6vThMZ5+DrGpkldicSBkvAME0JLIbsnAlltCUCWbkwGOY7PqKo3U/JLJVhHsATE86YVsu18WhtYbSoDrVjHsxFOcwPsv71r+6GQo3oe/oyhMy2+1TQhDBd2f0W4x8Apj7FfCWtxq4MkS9BiD5FGoio0bpVow3MDEv/CZr5Gi2wPS0xK6JNGg4X2m28OAYKN0lbJdBeuFlceJTwkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\ln\\left|\\frac{1}{\\sqrt{3}}x+\\sqrt{\\frac{3+x^{2}}{3}}\\right|" }, { "type": "step", "result": "=\\ln\\left|\\frac{1}{\\sqrt{3}}x+\\sqrt{\\frac{1}{3}\\left(3+x^{2}\\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": "=\\ln\\left|\\frac{1}{\\sqrt{3}}x+\\sqrt{\\frac{1}{3}\\left(3+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\\left|\\frac{1}{\\sqrt{3}}x+\\sqrt{\\frac{1}{3}(3+x^{2})}\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }