integral of (x^3)/(6-x^4)

{ "query": { "display": "$$\\int\\:\\frac{x^{3}}{6-x^{4}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{x^{3}}{6-x^{4}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{1}{4}\\ln\\left|6-x^{4}\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{x^{3}}{6-x^{4}}dx=-\\frac{1}{4}\\ln\\left|6-x^{4}\\right|+C$$", "input": "\\int\\:\\frac{x^{3}}{6-x^{4}}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{x^{3}}{6-x^{4}}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: $$u=6-x^{4}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=-4x^{3}$$", "input": "\\frac{d}{dx}\\left(6-x^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(6\\right)-\\frac{d}{dx}\\left(x^{4}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(6\\right)=0$$", "input": "\\frac{d}{dx}\\left(6\\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+idP5vLVrjUll6eMdYg8p7Gq8hcikAAMclWLaxZJJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtyoiJomQLyoKTDK4FJPEzd" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{4}\\right)=4x^{3}$$", "input": "\\frac{d}{dx}\\left(x^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4x^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4x^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYodvM0LC1fPom5KofzCj+6qk3hxk9aCfAWodBRxXgUexGgZz1CFzF7HTa4VF2uoRHv8//6/nV5O4fb8Xgwi7maqO95Is7YBdcRqXmTH8euc2JrcUvyUpj++aXrGYPlvDVw==" } }, { "type": "step", "result": "=0-4x^{3}" }, { "type": "step", "primary": "Simplify", "result": "=-4x^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=-4x^{3}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\left(-\\frac{1}{4x^{3}}\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{x^{3}}{u}\\left(-\\frac{1}{4x^{3}}\\right)du" }, { "type": "interim", "title": "Simplify $$\\frac{x^{3}}{u}\\left(-\\frac{1}{4x^{3}}\\right):{\\quad}-\\frac{1}{4u}$$", "input": "\\frac{x^{3}}{u}\\left(-\\frac{1}{4x^{3}}\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{x^{3}}{u}\\cdot\\:\\frac{1}{4x^{3}}" }, { "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\\cdot\\:4x^{3}}" }, { "type": "step", "primary": "Cancel the common factor: $$x^{3}$$", "result": "=-\\frac{1}{u\\cdot\\:4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{4u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+Cej/R+l5g8p9nSYVKvjunfBZx5IqsctXcVP4p+vJDx+qPBBXW0OBa8HfjFPmx5EoAIezcf2HGXjCQp0SyS6as7d3vw7eocUrI0TgQGIqC/iKCYpG7oXh2OSQAGUJjjefUZDHRRVkrDOQpTZTNW6opSBv6izheLVUKQ/emokAUyEKC4SmDfgdaB0h/PWlULm7CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{4u}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}\\cdot\\:\\int\\:\\frac{1}{u}du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u}du=\\ln\\left(\\left|u\\right|\\right)$$", "result": "=-\\frac{1}{4}\\ln\\left|u\\right|" }, { "type": "step", "primary": "Substitute back $$u=6-x^{4}$$", "result": "=-\\frac{1}{4}\\ln\\left|6-x^{4}\\right|" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{1}{4}\\ln\\left|6-x^{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}{4}\\ln\\left|6-x^{4}\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }