integral of sqrt(t^3+1)

{ "query": { "display": "$$\\int\\:\\sqrt{t^{3}+1}dt$$", "symbolab_question": "BIG_OPERATOR#\\int \\sqrt{t^{3}+1}dt" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{2(t^{4}+\\sqrt[6]{-1}\\cdot 3^{\\frac{3}{4}}\\F(\\sqrt[3]{-1}\\vert \\arcsin(\\frac{1}{\\sqrt[4]{3}}\\sqrt{-(-1)^{\\frac{5}{6}}(t+1)}))\\sqrt{-\\sqrt[6]{-1}(t+1)}\\sqrt{t^{2}-t+1}+t)}{5\\sqrt{t^{3}+1}}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\sqrt{t^{3}+1}dt=\\frac{2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{1}{\\sqrt[4]{3}}\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}+t\\right)}{5\\sqrt{t^{3}+1}}+C$$", "input": "\\int\\:\\sqrt{t^{3}+1}dt", "steps": [ { "type": "step", "primary": "Use the nonelementary integral: $$\\int\\:\\sqrt{x^{3}+1}dx=\\frac{2\\left(x^{4}+\\sqrt[6]{-1}3^{\\frac{3}{4}}\\sqrt{-\\sqrt[6]{-1}\\left(x+\\left(-1\\right)^{\\frac{2}{3}}\\right)}\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}x^{2}+\\sqrt[3]{-1}x+1}\\F\\left(\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(x+1\\right)}}{\\sqrt[4]{3}}\\right)\\vert\\:\\sqrt[3]{-1}\\right)+x\\right)}{5\\sqrt{x^{3}+1}}$$", "result": "=\\frac{2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\sqrt{-\\sqrt[6]{-1}\\left(t+\\left(-1\\right)^{\\frac{2}{3}}\\right)}\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}\\F\\left(\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\vert\\:\\sqrt[3]{-1}\\right)+t\\right)}{5\\sqrt{t^{3}+1}}" }, { "type": "interim", "title": "Simplify $$\\frac{2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\sqrt{-\\sqrt[6]{-1}\\left(t+\\left(-1\\right)^{\\frac{2}{3}}\\right)}\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}\\F\\left(\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\vert\\:\\sqrt[3]{-1}\\right)+t\\right)}{5\\sqrt{t^{3}+1}}:{\\quad}\\frac{2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{1}{\\sqrt[4]{3}}\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}+t\\right)}{5\\sqrt{t^{3}+1}}$$", "input": "\\frac{2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\sqrt{-\\sqrt[6]{-1}\\left(t+\\left(-1\\right)^{\\frac{2}{3}}\\right)}\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}\\F\\left(\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\vert\\:\\sqrt[3]{-1}\\right)+t\\right)}{5\\sqrt{t^{3}+1}}", "result": "=\\frac{2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{1}{\\sqrt[4]{3}}\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}+t\\right)}{5\\sqrt{t^{3}+1}}", "steps": [ { "type": "interim", "title": "$$2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\sqrt{-\\sqrt[6]{-1}\\left(t+\\left(-1\\right)^{\\frac{2}{3}}\\right)}\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}\\F\\left(\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\vert\\:\\sqrt[3]{-1}\\right)+t\\right)=2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}+t\\right)$$", "input": "2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\sqrt{-\\sqrt[6]{-1}\\left(t+\\left(-1\\right)^{\\frac{2}{3}}\\right)}\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}\\F\\left(\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\vert\\:\\sqrt[3]{-1}\\right)+t\\right)", "steps": [ { "type": "interim", "title": "$$\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\sqrt{-\\sqrt[6]{-1}\\left(t+\\left(-1\\right)^{\\frac{2}{3}}\\right)}\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}\\F\\left(\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\vert\\:\\sqrt[3]{-1}\\right)=\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}$$", "input": "\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\sqrt{-\\sqrt[6]{-1}\\left(t+\\left(-1\\right)^{\\frac{2}{3}}\\right)}\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}\\F\\left(\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\vert\\:\\sqrt[3]{-1}\\right)", "steps": [ { "type": "interim", "title": "$$\\sqrt{-\\sqrt[6]{-1}\\left(t+\\left(-1\\right)^{\\frac{2}{3}}\\right)}=\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}$$", "input": "\\sqrt{-\\sqrt[6]{-1}\\left(t+\\left(-1\\right)^{\\frac{2}{3}}\\right)}", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{\\frac{2}{3}}=1$$", "input": "\\left(-1\\right)^{\\frac{2}{3}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{\\frac{m}{n}}=\\left(a^{\\frac{1}{n}}\\right)^{m}$$", "secondary": [ "$$\\left(-1\\right)^{\\frac{2}{3}}=\\left(\\sqrt[3]{-1}\\right)^{2}$$" ], "result": "=\\left(\\sqrt[3]{-1}\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$\\sqrt[n]{-1}=-1,\\:$$if $$n$$ is odd", "secondary": [ "$$\\sqrt[3]{-1}=-1$$" ], "result": "=\\left(-1\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}=1$$" ], "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7g+dgv8bkUR6cGZTiWs7lgpc2CNzl3ejcl8+MKI7L6xxwkKGJWEPFPk38sdJMsyPIeqXfySbC6vm4UawE43QWXRsGar6AWJ3Y+vtwnYNNbMLZljS8Y2eU2XcV9yOaQM4R" } }, { "type": "step", "result": "=\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hCd+XOlWa1l7K6wVDIx9tHVM3M9876xO2HE1jVIQQsiFmFSw8FmiTCIoLDDc8XqyLTrWWMFI8l4Q07DZ5+hJa96vfCX/Yj0YmqBs3FRBhhL+w45/QVBZZgOGBYPWhiHut6tT2KeNVRpQMHz15Mv/h9fprFf8DlN5FCdeTM91rYi8gLb7RHL/+kBh/nd9INrzHJvzmVSpEDk8eIUvfjX6KmdjIn+Dsw4Vo8Yj2gjA9R46n//7zyXiup4CB0t5mn7wsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "=\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}" }, { "type": "interim", "title": "$$\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}=\\sqrt{t^{2}-t+1}$$", "input": "\\sqrt{\\left(-1\\right)^{\\frac{2}{3}}t^{2}+\\sqrt[3]{-1}t+1}", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{\\frac{2}{3}}t^{2}=t^{2}$$", "input": "\\left(-1\\right)^{\\frac{2}{3}}t^{2}", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{\\frac{2}{3}}=1$$", "input": "\\left(-1\\right)^{\\frac{2}{3}}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{\\frac{m}{n}}=\\left(a^{\\frac{1}{n}}\\right)^{m}$$", "secondary": [ "$$\\left(-1\\right)^{\\frac{2}{3}}=\\left(\\sqrt[3]{-1}\\right)^{2}$$" ], "result": "=\\left(\\sqrt[3]{-1}\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "$$\\sqrt[n]{-1}=-1,\\:$$if $$n$$ is odd", "secondary": [ "$$\\sqrt[3]{-1}=-1$$" ], "result": "=\\left(-1\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}=1$$" ], "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7g+dgv8bkUR6cGZTiWs7lgpc2CNzl3ejcl8+MKI7L6xxwkKGJWEPFPk38sdJMsyPIeqXfySbC6vm4UawE43QWXRsGar6AWJ3Y+vtwnYNNbMLZljS8Y2eU2XcV9yOaQM4R" } }, { "type": "step", "result": "=1\\cdot\\:t^{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:t^{2}=t^{2}$$", "result": "=t^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7g+dgv8bkUR6cGZTiWs7lgnKxg/lIopzPOT/7tXlwgE3NGoPE9TME3q+OPmgkv2RQBTziRcNZ1rL4dBGWK+4pYNFL/oxAMBAORV0NyMFZ/zM0r/NCt7v9aLVtzYb6puTWmV2wc1EMOlDR65gTM/wZCQ==" } }, { "type": "interim", "title": "$$\\sqrt[3]{-1}t=-t$$", "input": "\\sqrt[3]{-1}t", "steps": [ { "type": "interim", "title": "$$\\sqrt[3]{-1}=-1$$", "input": "\\sqrt[3]{-1}", "steps": [ { "type": "step", "primary": "$$\\sqrt[n]{-1}=-1,\\:$$if $$n$$ is odd", "secondary": [ "$$\\sqrt[3]{-1}=-1$$" ], "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76H1xoeptwwOonZRztLDv7i061ljBSPJeENOw2efoSWuixCfF7o9L2hRx9Shm0jmvo3oe/oyhMy2+1TQhDBd2f8qdr6Xp5CVvDj5nIWgDseQkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-1\\cdot\\:t" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:t=t$$", "result": "=-t" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X+sVuGGeXB30mrk2Herb3ACWKUbvV6WK3fDUgFtg3Q+cYGIaY4d0bYFQhlqEGdbOZEt3ZXAiqUE0HIXrrrezJIRD8oyyyKN/kakEFyk8fRmwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\sqrt{t^{2}-t+1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7P8ePe6rnXSqqdLoLhLCBWHnvMAIWjSNsy2z+L4rlMm9jtmfvOQDyvt7P1JSCctrT/KtF2sjFUSIoehkmXzUmKKORWLXkjysF58uLgjK3bCtt7scAXBkGnkc+ZiTAn8Px/z//r+dXk7h9vxeDCLuZqlMKJG5g56sxCJQ58q6Kv07Vy+QCk2N3+4whiGBiIpbtwOUiJgCSMH5f8C+rfd//EApD0jYTFVMh1mvbaIuZj2A5GGax8Z8y9ha0TxvXfkRj" } }, { "type": "step", "result": "=\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "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" } }, { "type": "step", "result": "=2\\left(t^{4}+t+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "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" } }, { "type": "step", "result": "=\\frac{2\\left(t^{4}+t+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}}{\\sqrt[4]{3}}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}\\right)}{5\\sqrt{t^{3}+1}}" }, { "type": "step", "result": "=\\frac{2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}\\F\\left(\\arcsin\\left(\\frac{1}{\\sqrt[4]{3}}\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}\\right)\\vert\\:\\sqrt[3]{-1}\\right)+t\\right)}{5\\sqrt{t^{3}+1}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{2\\left(t^{4}+\\sqrt[6]{-1}\\cdot\\:3^{\\frac{3}{4}}\\F\\left(\\sqrt[3]{-1}\\vert\\:\\arcsin\\left(\\frac{1}{\\sqrt[4]{3}}\\sqrt{-\\left(-1\\right)^{\\frac{5}{6}}\\left(t+1\\right)}\\right)\\right)\\sqrt{-\\sqrt[6]{-1}\\left(t+1\\right)}\\sqrt{t^{2}-t+1}+t\\right)}{5\\sqrt{t^{3}+1}}+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" } }, "meta": { "showVerify": true } }