integral of 12(y^4+4y^2+8)^2(y^3+2y)

{ "query": { "display": "$$\\int\\:12\\left(y^{4}+4y^{2}+8\\right)^{2}\\left(y^{3}+2y\\right)dy$$", "symbolab_question": "BIG_OPERATOR#\\int 12(y^{4}+4y^{2}+8)^{2}(y^{3}+2y)dy" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "(y^{4}+4y^{2}+8)^{3}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:12\\left(y^{4}+4y^{2}+8\\right)^{2}\\left(y^{3}+2y\\right)dy=\\left(y^{4}+4y^{2}+8\\right)^{3}+C$$", "input": "\\int\\:12\\left(y^{4}+4y^{2}+8\\right)^{2}\\left(y^{3}+2y\\right)dy", "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": "=12\\cdot\\:\\int\\:\\left(y^{4}+4y^{2}+8\\right)^{2}\\left(y^{3}+2y\\right)dy" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\left(y^{4}+4y^{2}+8\\right)^{2}\\left(y^{3}+2y\\right)dy", "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: $$u=y^{4}+4y^{2}+8$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dy}=4y^{3}+8y$$", "input": "\\frac{d}{dy}\\left(y^{4}+4y^{2}+8\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dy}\\left(y^{4}\\right)+\\frac{d}{dy}\\left(4y^{2}\\right)+\\frac{d}{dy}\\left(8\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dy}\\left(y^{4}\\right)=4y^{3}$$", "input": "\\frac{d}{dy}\\left(y^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4y^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4y^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkxhypVUiG3atzXmTPYRdMak3hxk9aCfAWodBRxXgUexg78CTHCDA/lpFNIL9ZtcH/8//6/nV5O4fb8Xgwi7maozYMeOxPk/lnQlt7VFBYbY90eL1Lx58KCcIoFHnNQcCw==" } }, { "type": "interim", "title": "$$\\frac{d}{dy}\\left(4y^{2}\\right)=8y$$", "input": "\\frac{d}{dy}\\left(4y^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dy}\\left(y^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4\\cdot\\:2y^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=8y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYrqt0YuaSL2Rm1UYrS/xAkGTdaV09PMxEKZ9FieghTFwZ2fGfNjpdOxZtpqB42U58KN6Hv6MoTMtvtU0IQwXdn/ap0oHImbv6f5GTOTwYbXNksmWoz/3QD0mF0Vr38VpIg==" } }, { "type": "interim", "title": "$$\\frac{d}{dy}\\left(8\\right)=0$$", "input": "\\frac{d}{dy}\\left(8\\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+idP5vLVrjUll6eMdYkxl41yJ3wnOwbaINEo5sfFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtpgS80+Yza2KfnoIr71kZr" } }, { "type": "step", "result": "=4y^{3}+8y+0" }, { "type": "step", "primary": "Simplify", "result": "=4y^{3}+8y", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=4y^{3}+8ydy$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dy=\\frac{1}{4y^{3}+8y}du$$" }, { "type": "step", "result": "=\\int\\:u^{2}\\left(y^{3}+2y\\right)\\frac{1}{4y^{3}+8y}du" }, { "type": "interim", "title": "Simplify $$u^{2}\\left(y^{3}+2y\\right)\\frac{1}{4y^{3}+8y}:{\\quad}\\frac{u^{2}}{4}$$", "input": "u^{2}\\left(y^{3}+2y\\right)\\frac{1}{4y^{3}+8y}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:u^{2}\\left(y^{3}+2y\\right)}{4y^{3}+8y}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:u^{2}=u^{2}$$", "result": "=\\frac{u^{2}\\left(y^{3}+2y\\right)}{4y^{3}+8y}" }, { "type": "interim", "title": "Factor $$y^{3}+2y:{\\quad}y\\left(y^{2}+2\\right)$$", "input": "y^{3}+2y", "result": "=\\frac{u^{2}y\\left(y^{2}+2\\right)}{4y^{3}+8y}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$y^{3}=yy^{2}$$" ], "result": "=yy^{2}+2y", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$y$$", "result": "=y\\left(y^{2}+2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$4y^{3}+8y:{\\quad}4y\\left(y^{2}+2\\right)$$", "input": "4y^{3}+8y", "result": "=\\frac{u^{2}y\\left(y^{2}+2\\right)}{4y\\left(y^{2}+2\\right)}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$y^{3}=yy^{2}$$" ], "result": "=4yy^{2}+8y", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Rewrite as", "result": "=4yy^{2}+2\\cdot\\:4y" }, { "type": "step", "primary": "Factor out common term $$4y$$", "result": "=4y\\left(y^{2}+2\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Cancel $$\\frac{u^{2}y\\left(y^{2}+2\\right)}{4y\\left(y^{2}+2\\right)}:{\\quad}\\frac{u^{2}}{4}$$", "input": "\\frac{u^{2}y\\left(y^{2}+2\\right)}{4y\\left(y^{2}+2\\right)}", "result": "=\\frac{u^{2}}{4}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$y$$", "result": "=\\frac{u^{2}\\left(y^{2}+2\\right)}{4\\left(y^{2}+2\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$y^{2}+2$$", "result": "=\\frac{u^{2}}{4}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnAwSLBELR8i9m6G4P+CisrAbMFVLe2Vt4pfIAwMbkqLVdNK6b/PmQukzNdLVQkh7GCeK3dtjkfH1uaY9WqMYsCuirxfJj7yzhlUWCbvijZZ1sD7NfhsPe7eDHrmjY0mE2J/JAWPJQkixCwxTVklZhNwMEiwRC0fIvZuhuD/gorK63VPOevF/u6/soDtZLyG8SS3daIZHtloJpe/PvtsyNI=" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{u^{2}}{4}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7y4NDkPMEyDrWlXR5OwenVlB/rfq1RcQ1BqqvOJn9yVqHI5S0StY1FdtOqqOPr0Te8kZykkSQR7JxF6tcEv88MO4fClF/YTdtcCRNeh/zX2cdnL+ucZY+BVjv4wL+wJtQuD2/7SwQCayxtHsn6FD9znvbBmbuQNTF0TphKZ8RuvaR7hM6KADkFDngi1m57r+mR5n4zu7ICtqiQHGrVYWXPk=" } }, { "type": "step", "result": "=12\\cdot\\:\\int\\:\\frac{u^{2}}{4}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": "=12\\cdot\\:\\frac{1}{4}\\cdot\\:\\int\\:u^{2}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:u^{2}du", "result": "=12\\cdot\\:\\frac{1}{4}\\cdot\\:\\frac{u^{3}}{3}", "steps": [ { "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 $$\\frac{u^{2+1}}{2+1}:{\\quad}\\frac{u^{3}}{3}$$", "input": "\\frac{u^{2+1}}{2+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=\\frac{u^{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{u^{3}}{3}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7ztMnkJqJUs4Qp2osi+oqZ+o/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ofEE1c1KRzl1GtEOEJ8QtOggQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "step", "primary": "Substitute back $$u=y^{4}+4y^{2}+8$$", "result": "=12\\cdot\\:\\frac{1}{4}\\cdot\\:\\frac{\\left(y^{4}+4y^{2}+8\\right)^{3}}{3}" }, { "type": "interim", "title": "Simplify $$12\\cdot\\:\\frac{1}{4}\\cdot\\:\\frac{\\left(y^{4}+4y^{2}+8\\right)^{3}}{3}:{\\quad}\\left(y^{4}+4y^{2}+8\\right)^{3}$$", "input": "12\\cdot\\:\\frac{1}{4}\\cdot\\:\\frac{\\left(y^{4}+4y^{2}+8\\right)^{3}}{3}", "result": "=\\left(y^{4}+4y^{2}+8\\right)^{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=\\frac{1\\cdot\\:\\left(y^{4}+4y^{2}+8\\right)^{3}\\cdot\\:12}{4\\cdot\\:3}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{12\\left(y^{4}+4y^{2}+8\\right)^{3}}{12}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{12}{12}=1$$", "result": "=\\left(y^{4}+4y^{2}+8\\right)^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X9Re+PckSoemCIRQQUfDdRKeGTTEZItbVonaoPZxnfW4BKkWMzJDVj+NkZEF+9jUzDkaBtFcVK1CC0iFZmmWlN6GQqufR6tr2vPxOUv7H++NYxOQPujgok9wizXRQ0bKpkSO8SEQ2VyMAzM4ybVPNaN6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwgmYMUHWuuN2/A63BuN3hQoikJfK6myGFkAjFYK+SmYXZrfQkkunM4TFS4yddbdBo2bdn7epyh2nsXoTCgpWwSyQ==" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\left(y^{4}+4y^{2}+8\\right)^{3}+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": "y", "plotRequest": "y=(y^{4}+4y^{2}+8)^{3}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }