integral of (x^3)/((x^2+4)^2)

{ "query": { "display": "$$\\int\\:\\frac{x^{3}}{\\left(x^{2}+4\\right)^{2}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{x^{3}}{(x^{2}+4)^{2}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{1}{2}(\\ln\\left|x^{2}+4\\right|+\\frac{4}{x^{2}+4})+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{x^{3}}{\\left(x^{2}+4\\right)^{2}}dx=\\frac{1}{2}\\left(\\ln\\left|x^{2}+4\\right|+\\frac{4}{x^{2}+4}\\right)+C$$", "input": "\\int\\:\\frac{x^{3}}{\\left(x^{2}+4\\right)^{2}}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{x^{3}}{\\left(x^{2}+4\\right)^{2}}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: $$u=x^{2}+4$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+4\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(4\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4\\right)=0$$", "input": "\\frac{d}{dx}\\left(4\\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+idP5vLVrjUll6eMdYjVwwDW+HeFUFiKZ8J+l8XpJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTt4/nDM7CraQVY2V0O4nKcI" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2xdx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{2x}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{x^{3}}{u^{2}}\\cdot\\:\\frac{1}{2x}du" }, { "type": "interim", "title": "Simplify $$\\frac{x^{3}}{u^{2}}\\cdot\\:\\frac{1}{2x}:{\\quad}\\frac{x^{2}}{2u^{2}}$$", "input": "\\frac{x^{3}}{u^{2}}\\cdot\\:\\frac{1}{2x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{x^{3}\\cdot\\:1}{u^{2}\\cdot\\:2x}" }, { "type": "step", "primary": "Multiply: $$x^{3}\\cdot\\:1=x^{3}$$", "result": "=\\frac{x^{3}}{2xu^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{x^{2}}{2u^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{x^{2}}{2u^{2}}du" }, { "type": "interim", "title": "$$u=x^{2}+4\\quad\\Rightarrow\\quad\\:x^{2}=u-4$$", "input": "x^{2}+4=u", "steps": [ { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "x^{2}+4=u", "result": "x^{2}=u-4", "steps": [ { "type": "step", "primary": "Subtract $$4$$ from both sides", "result": "x^{2}+4-4=u-4" }, { "type": "step", "primary": "Simplify", "result": "x^{2}=u-4" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\frac{u-4}{2u^{2}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74sQsa/I7yO3h7kUeoexLgmZ+f6NaqEuuMIEEWawncC4pN4cZPWgnwFqHQUcV4FHsfBLh5j/jJcd1Frv9s/1xSw0pWMfsJc1e/Z0+a/wFZqiVdawfdGtW4aUXHrK9DlWLZdfaS+FZXUniz7lchSYJLRFKk3fejFkyiOiq9iG9IkAWXoEtoMIfKav4MDX8Gg0WwMDU99yPX7ThnZcoXB515Q=" } }, { "type": "step", "result": "=\\int\\:\\frac{u-4}{2u^{2}}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}{2}\\cdot\\:\\int\\:\\frac{u-4}{u^{2}}du" }, { "type": "interim", "title": "Expand $$\\frac{u-4}{u^{2}}:{\\quad}\\frac{1}{u}-\\frac{4}{u^{2}}$$", "input": "\\frac{u-4}{u^{2}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{u-4}{u^{2}}=\\frac{u}{u^{2}}-\\frac{4}{u^{2}}$$" ], "result": "=\\frac{u}{u^{2}}-\\frac{4}{u^{2}}" }, { "type": "interim", "title": "Cancel $$\\frac{u}{u^{2}}:{\\quad}\\frac{1}{u}$$", "input": "\\frac{u}{u^{2}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$u$$", "result": "=\\frac{1}{u}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYvF4q53Fy7hnUeqV16eRQebNGoPE9TME3q+OPmgkv2RQoOCh4VQCzJ+S9FmwraXOdTpsURahcgs/m/opDzXaTtppUOtUXFOWOYuKTxbUMEVbsmnQmDV6mP8x67mJBW8kOBZFQ8OchBdgxMOJrEMyk3s=" } }, { "type": "step", "result": "=\\frac{1}{u}-\\frac{4}{u^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PvcFqgR4aFJJEm/lxOeQbSkWnF/8WxWzZ96351/38mDMwViaLUXkeD+JukROhWdjb8di2xBlql84x+NhWh4C88Bc9CJZ7ZWx0XyP+ln1BAyjeh7+jKEzLb7VNCEMF3Z/TzUpvge51uP/vAYJhDDKq2Zc1F/WihaufPh26pVwc2SJqVxX90jlMfh9fKn6dzC4" } }, { "type": "step", "result": "=\\frac{1}{2}\\cdot\\:\\int\\:\\frac{1}{u}-\\frac{4}{u^{2}}du" }, { "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": "=\\frac{1}{2}\\left(\\int\\:\\frac{1}{u}du-\\int\\:\\frac{4}{u^{2}}du\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{u}du=\\ln\\left|u\\right|$$", "input": "\\int\\:\\frac{1}{u}du", "steps": [ { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u}du=\\ln\\left(\\left|u\\right|\\right)$$", "result": "=\\ln\\left|u\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7xYpRYRjORiCJkHaCCs/U61fRzuAubUUJYHDgzAJMdz/8+LBiPyAP34u+4MuPtfQQvGeKNchbzBJJgE/Z7UYTEUZH/yLEwzXgrACX51mM8dC3KorEA/Q0CbMb/fBf5r55LRNvOTr4fZAVVYGYBAkbV86k5tT+RaEjrsoFUO/AsBKtTI+06tguYsUGJG/KKWkTQ==" } }, { "type": "interim", "title": "$$\\int\\:\\frac{4}{u^{2}}du=-\\frac{4}{u}$$", "input": "\\int\\:\\frac{4}{u^{2}}du", "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": "=4\\cdot\\:\\int\\:\\frac{1}{u^{2}}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\frac{1}{u^{2}}du", "result": "=4\\left(-\\frac{1}{u}\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{u^{2}}=u^{-2}$$" ], "result": "=\\int\\:u^{-2}du", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{u^{-2+1}}{-2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{-2+1}}{-2+1}:{\\quad}-\\frac{1}{u}$$", "input": "\\frac{u^{-2+1}}{-2+1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=\\frac{u^{-1}}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{u^{-1}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-u^{-1}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-1}=\\frac{1}{a}$$", "result": "=-\\frac{1}{u}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-\\frac{1}{u}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/RSr02Agv0MR/qV7Nm+eMMy4+rY5ULRUEksemusM4Yyrrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx4OQzbEnsuafNY3nX9QxDlJ1HXTSqqQEjS1gpf6I+JyHQS4M5VpC8qh+oehjmM1qmweKkh+28FiXwy+Vsz8xLQiialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$4\\left(-\\frac{1}{u}\\right):{\\quad}-\\frac{4}{u}$$", "input": "4\\left(-\\frac{1}{u}\\right)", "result": "=-\\frac{4}{u}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-4\\cdot\\:\\frac{1}{u}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:4}{u}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:4=4$$", "result": "=-\\frac{4}{u}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TCI2hyj1sQeh4jCd0Mjj5+Yxsyt+/vCP3mLOgEDBrSd5tMpJTBBccUWkSyvMe1SpRZBofyAsXcBnk1L4ZLUWu+9sGZu5A1MXROmEpnxG69qt4movs/wPf6iZqsxc5tOd8Dc3za0XlzspxmUcVz8AqL8yD3hLQ33B7/8/LpbPE3o=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{1}{2}\\left(\\ln\\left|u\\right|-\\left(-\\frac{4}{u}\\right)\\right)" }, { "type": "step", "primary": "Substitute back $$u=x^{2}+4$$", "result": "=\\frac{1}{2}\\left(\\ln\\left|x^{2}+4\\right|-\\left(-\\frac{4}{x^{2}+4}\\right)\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{2}\\left(\\ln\\left|x^{2}+4\\right|+\\frac{4}{x^{2}+4}\\right)", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{1}{2}\\left(\\ln\\left|x^{2}+4\\right|+\\frac{4}{x^{2}+4}\\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|x^{2}+4\\right|+\\frac{4}{x^{2}+4})+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }