integral of (6e^u-u^3(sqrt(u)+1))

{ "query": { "display": "$$\\int\\:\\left(6e^{u}-u^{3}\\left(\\sqrt{u}+1\\right)\\right)du$$", "symbolab_question": "BIG_OPERATOR#\\int (6e^{u}-u^{3}(\\sqrt{u}+1))du" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "6e^{u}-\\frac{2}{9}u^{\\frac{9}{2}}-\\frac{u^{4}}{4}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\left(6e^{u}-u^{3}\\left(\\sqrt{u}+1\\right)\\right)du=6e^{u}-\\frac{2}{9}u^{\\frac{9}{2}}-\\frac{u^{4}}{4}+C$$", "input": "\\int\\:6e^{u}-u^{3}\\left(\\sqrt{u}+1\\right)du", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:6e^{u}-u^{3}\\left(\\sqrt{u}+1\\right)du", "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: $$v=e^{u}$$" ] }, { "type": "interim", "title": "$$\\frac{dv}{du}=e^{u}$$", "input": "\\frac{d}{du}\\left(e^{u}\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(e^{u}\\right)=e^{u}$$", "result": "=e^{u}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqCr3EWRZw3L4+rHTTdVG0Ok3hxk9aCfAWodBRxXgUexwx+RE9MtjN5hKMwTI7fffj/L0MoYg+CUn6oyL3EO7YogoXEKjQiSEApQd5DWfyV6oqL0GmPYnsUjDz0ibjuXcg==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=e^{u}du$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{1}{e^{u}}dv$$" }, { "type": "step", "result": "=\\int\\:\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)\\frac{1}{e^{u}}dv" }, { "type": "step", "primary": "$$v=e^{u}$$", "result": "=\\int\\:\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)\\frac{1}{v}dv" }, { "type": "interim", "title": "Simplify $$\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)\\frac{1}{v}:{\\quad}\\frac{6v-u^{3}\\left(\\sqrt{u}+1\\right)}{v}$$", "input": "\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)\\frac{1}{v}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)}{v}" }, { "type": "interim", "title": "$$1\\cdot\\:\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)=6v-u^{3}\\left(\\sqrt{u}+1\\right)$$", "input": "1\\cdot\\:\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)=\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)$$", "result": "=\\left(6v-u^{3}\\left(\\sqrt{u}+1\\right)\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=6v-u^{3}\\left(\\sqrt{u}+1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72LPaMs5UWBFirRwV4yoDGRnejcBNXgFXc++0KkR3ZnfTLx8mOdHYVzxX643JqKFI1Zm+16sW8yUrLyG3kcD8QHO4vI0xqGEClwmJ8Kj65/J9MYWmeKc69KU0DwRz3/ycWrTRYnlYeaMFAJhYa6w1O1/Vq63zSUV5ngwax4P7QyPYE6I3WRBHohYMQMgbY5bTQ7yln6QvYVlclgXvTgE41A==" } }, { "type": "step", "result": "=\\frac{6v-u^{3}\\left(\\sqrt{u}+1\\right)}{v}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{6v-u^{3}\\left(\\sqrt{u}+1\\right)}{v}dv" }, { "type": "interim", "title": "$$v=e^{u}\\quad\\Rightarrow\\quad\\:u=\\ln\\left(v\\right)$$", "input": "e^{u}=v", "steps": [ { "type": "interim", "title": "Apply exponent rules", "input": "e^{u}=v", "result": "u=\\ln\\left(v\\right)", "steps": [ { "type": "step", "primary": "If $$f\\left(x\\right)=g\\left(x\\right)$$, then $$\\ln\\left(f\\left(x\\right)\\right)=\\ln\\left(g\\left(x\\right)\\right)$$", "result": "\\ln\\left(e^{u}\\right)=\\ln\\left(v\\right)" }, { "type": "step", "primary": "Apply log rule: $$\\ln\\left(e^a\\right)=a$$", "secondary": [ "$$\\ln\\left(e^{u}\\right)=u$$" ], "result": "u=\\ln\\left(v\\right)", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Apply Exp Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77JjsBWWVfWP0Aur62hK4oH6c16bS4z9BC4Kh6bTmYjahHeJ1xBi321LjY4vsXC9em3FAiPzxVy0umodhDNEdMgI+pIcEOdT3dYPD7SQwdX9kS3dlcCKpQTQcheuut7Mk5G8dMNkjSttSrmZL423XYQkTRBwR2IXB9bWSAefdvcw=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\frac{6v-\\ln^{3}\\left(v\\right)\\left(\\sqrt{\\ln\\left(v\\right)}+1\\right)}{v}dv" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74pSuxLkIhG/tLook5xJAzJVQYaOkhiZPQrOCxk6tyJ9LI71+ylVDvwHghUiHeEt93Gam9ro5jdhRGQuTP5rGCB4l9+XUP4dNZSL8WpmbbeTmg5o2p0nQSJ3i8HrTWzQZpWPuzJ6e+xnddXsvq7+f7J2ZZY1+Cpq83pfXtlPh0sD15B902njgUxJxt4erO51ZgS4M5VpC8qh+oehjmM1qmxw3obfALRo06lfcYy06xaEialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "result": "=\\int\\:\\frac{6v-\\ln^{3}\\left(v\\right)\\left(\\sqrt{\\ln\\left(v\\right)}+1\\right)}{v}dv" }, { "type": "interim", "title": "Expand $$\\frac{6v-\\ln^{3}\\left(v\\right)\\left(\\sqrt{\\ln\\left(v\\right)}+1\\right)}{v}:{\\quad}6-\\frac{\\ln^{\\frac{7}{2}}\\left(v\\right)}{v}-\\frac{\\ln^{3}\\left(v\\right)}{v}$$", "input": "\\frac{6v-\\ln^{3}\\left(v\\right)\\left(\\sqrt{\\ln\\left(v\\right)}+1\\right)}{v}", "steps": [ { "type": "interim", "title": "Expand $$6v-\\ln^{3}\\left(v\\right)\\left(\\sqrt{\\ln\\left(v\\right)}+1\\right):{\\quad}6v-\\ln^{\\frac{7}{2}}\\left(v\\right)-\\ln^{3}\\left(v\\right)$$", "input": "6v-\\ln^{3}\\left(v\\right)\\left(\\sqrt{\\ln\\left(v\\right)}+1\\right)", "result": "=\\frac{6v-\\ln^{\\frac{7}{2}}\\left(v\\right)-\\ln^{3}\\left(v\\right)}{v}", "steps": [ { "type": "interim", "title": "Expand $$-\\ln^{3}\\left(v\\right)\\left(\\sqrt{\\ln\\left(v\\right)}+1\\right):{\\quad}-\\ln^{\\frac{7}{2}}\\left(v\\right)-\\ln^{3}\\left(v\\right)$$", "input": "-\\ln^{3}\\left(v\\right)\\left(\\sqrt{\\ln\\left(v\\right)}+1\\right)", "result": "=6v-\\ln^{\\frac{7}{2}}\\left(v\\right)-\\ln^{3}\\left(v\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=-\\ln^{3}\\left(v\\right),\\:b=\\sqrt{\\ln\\left(v\\right)},\\:c=1$$" ], "result": "=-\\ln^{3}\\left(v\\right)\\sqrt{\\ln\\left(v\\right)}+\\left(-\\ln^{3}\\left(v\\right)\\right)\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=-\\ln^{3}\\left(v\\right)\\sqrt{\\ln\\left(v\\right)}-1\\cdot\\:\\ln^{3}\\left(v\\right)" }, { "type": "interim", "title": "Simplify $$-\\ln^{3}\\left(v\\right)\\sqrt{\\ln\\left(v\\right)}-1\\cdot\\:\\ln^{3}\\left(v\\right):{\\quad}-\\ln^{\\frac{7}{2}}\\left(v\\right)-\\ln^{3}\\left(v\\right)$$", "input": "-\\ln^{3}\\left(v\\right)\\sqrt{\\ln\\left(v\\right)}-1\\cdot\\:\\ln^{3}\\left(v\\right)", "result": "=-\\ln^{\\frac{7}{2}}\\left(v\\right)-\\ln^{3}\\left(v\\right)", "steps": [ { "type": "interim", "title": "$$\\ln^{3}\\left(v\\right)\\sqrt{\\ln\\left(v\\right)}=\\ln^{\\frac{7}{2}}\\left(v\\right)$$", "input": "\\ln^{3}\\left(v\\right)\\sqrt{\\ln\\left(v\\right)}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$\\ln^{3}\\left(v\\right)\\sqrt{\\ln\\left(v\\right)}=\\:\\ln^{3}\\left(v\\right)\\ln^{\\frac{1}{2}}\\left(v\\right)=\\:\\ln^{3+\\frac{1}{2}}\\left(v\\right)$$" ], "result": "=\\ln^{3+\\frac{1}{2}}\\left(v\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Join $$3+\\frac{1}{2}:{\\quad}\\frac{7}{2}$$", "input": "3+\\frac{1}{2}", "result": "=\\ln^{\\frac{7}{2}}\\left(v\\right)", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$3=\\frac{3\\cdot\\:2}{2}$$", "result": "=\\frac{3\\cdot\\:2}{2}+\\frac{1}{2}" }, { "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{3\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$3\\cdot\\:2+1=7$$", "input": "3\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=6+1" }, { "type": "step", "primary": "Add the numbers: $$6+1=7$$", "result": "=7" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ksWDPsxIbGkmQroaF3uT7d6GQqufR6tr2vPxOUv7H+9MTg4418YnsnbKpNwPhLduVnEva1E6F4KI9o/Cnut2g24/Va/+G3hLP9WfhSb3X+c=" } }, { "type": "step", "result": "=\\frac{7}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OfvaoLyFzXr3swCdJRD2q45ZQMgjhhRPRxP/T93jeOEgJ/ZZA32ZInFBpDtxBfiKWyifSaKFbgRjkCBRl5QRuXFPnwb0emYk1CuWo4PRvT9kS3dlcCKpQTQcheuut7MkpoDp2/jj7zAAfbWLy3QKzDFMWNpOZNOjce0JiP09d4u8Zu5NwmCbAXJZbW5ACM5IsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "$$1\\cdot\\:\\ln^{3}\\left(v\\right)=\\ln^{3}\\left(v\\right)$$", "input": "1\\cdot\\:\\ln^{3}\\left(v\\right)", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\ln^{3}\\left(v\\right)=\\ln^{3}\\left(v\\right)$$", "result": "=\\ln^{3}\\left(v\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AF0YgndutBwz9qiWWLqHSnDq0lbpHPU/KUvt95E4qClwkKGJWEPFPk38sdJMsyPIa/WlvrgAu9zVbVtL3bC6BaBzIB6qdky9r1WITjaDuDSTl7nixrD+bJtZOj7S8PYZQFet09mJnKeyXhpi5Yvwtg==" } }, { "type": "step", "result": "=-\\ln^{\\frac{7}{2}}\\left(v\\right)-\\ln^{3}\\left(v\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zvE9D0E4K+66ZWreuyz/gay0ruzop1OE4gF+QisOmuHTLx8mOdHYVzxX643JqKFIwxWDXidEV9CzsGPnUu41zKOQXeKP8OOpZ+3HrX9l9BeedvuwcV8RTgVLFgPaHPKR7lvKtANBUJdQPS8f9+853PxtgDRvpXZFJM5ns6z004Mj+9sh0rp+4B3DEMYcU0bMrLSu7OinU4TiAX5CKw6a4eIASZeFjDtawNGt9P21GJY=" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eVu9xzpNXRs/HMUzU57ciEpj9ZzqEXANNijIB7ULKIk9oGLUBJeFe/crOKQ3lF41o5FYteSPKwXny4uCMrdsK5jBwMzuHQ4OMgx/BQiEe6n6gLOJWO2xPWn5Po64KqVDgWsjWHhiiuWPM+3OzhYVABJyf8zawtgEaDEKWrMLEzbcldX/2RG5s0yuBka/UFNbAgho6DJvW4r9HXAJ1Po1WcNgXi3FXcKyAeoGf5fQo6A=" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{6v-\\ln^{\\frac{7}{2}}\\left(v\\right)-\\ln^{3}\\left(v\\right)}{v}=\\frac{6v}{v}-\\frac{\\ln^{\\frac{7}{2}}\\left(v\\right)}{v}-\\frac{\\ln^{3}\\left(v\\right)}{v}$$" ], "result": "=\\frac{6v}{v}-\\frac{\\ln^{\\frac{7}{2}}\\left(v\\right)}{v}-\\frac{\\ln^{3}\\left(v\\right)}{v}" }, { "type": "interim", "title": "Cancel $$\\frac{6v}{v}:{\\quad}6$$", "input": "\\frac{6v}{v}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$v$$", "result": "=6" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgKCmn5GnafCemFcqc6uVo4JQJZuTAY5js+oqjdT8kslBdF3ow+Iomrch4PQ8hq4mPrIhudej3DKuXXR7wDVpFbvgb6hyCPgrjVBNg+ZKfCKialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "result": "=6-\\frac{\\ln^{\\frac{7}{2}}\\left(v\\right)}{v}-\\frac{\\ln^{3}\\left(v\\right)}{v}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jRkwJguNRw4aGYqhEjluV9C9EP+PWswnTfIMpYQyIkBqxNlrQ6YOkg/eho53pcajVdNK6b/PmQukzNdLVQkh7DpxbgadTS/V9pC1cS+fSCHog7gFZbgelZxEwRO/RMLWMiMPzaN6CX5rK+FzVpK/QK2a5CW+0jKjRUhZVIXPstnhm+qb7td31enZMokMe/HZEnJ/zNrC2ARoMQpaswsTNudns08Os7YGUjmZ6TIy5naVj7syenvsZ3XV7L6u/n+ydmWWNfgqavN6X17ZT4dLA/nGbzbnui4MSyLP0Cw/+jI=" } }, { "type": "step", "result": "=\\int\\:6-\\frac{\\ln^{\\frac{7}{2}}\\left(v\\right)}{v}-\\frac{\\ln^{3}\\left(v\\right)}{v}dv" }, { "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": "=\\int\\:6dv-\\int\\:\\frac{\\ln^{\\frac{7}{2}}\\left(v\\right)}{v}dv-\\int\\:\\frac{\\ln^{3}\\left(v\\right)}{v}dv" }, { "type": "interim", "title": "$$\\int\\:6dv=6v$$", "input": "\\int\\:6dv", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=6v" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{\\ln^{\\frac{7}{2}}\\left(v\\right)}{v}dv=\\frac{2}{9}\\ln^{\\frac{9}{2}}\\left(v\\right)$$", "input": "\\int\\:\\frac{\\ln^{\\frac{7}{2}}\\left(v\\right)}{v}dv", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\ln^{\\frac{7}{2}}\\left(v\\right)}{v}dv", "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: $$w=\\ln\\left(v\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{dw}{dv}=\\frac{1}{v}$$", "input": "\\frac{d}{dv}\\left(\\ln\\left(v\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dv}\\left(\\ln\\left(v\\right)\\right)=\\frac{1}{v}$$", "result": "=\\frac{1}{v}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqmzOQ63l6HV0QSyzcsb0SEcjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJitKzT0FSnQ1VzGfSlQmOzNTW26qciuyUBGXQExCUedYuGwmAwhCbJ/WP6zWvmSOUQgcJjF6wpuoCfd4QcipEuc=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dw=\\frac{1}{v}dv$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=vdw$$" }, { "type": "step", "result": "=\\int\\:\\frac{w^{\\frac{7}{2}}}{v}vdw" }, { "type": "interim", "title": "Simplify $$\\frac{w^{\\frac{7}{2}}}{v}v:{\\quad}w^{\\frac{7}{2}}$$", "input": "\\frac{w^{\\frac{7}{2}}}{v}v", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{w^{\\frac{7}{2}}v}{v}" }, { "type": "step", "primary": "Cancel the common factor: $$v$$", "result": "=w^{\\frac{7}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:w^{\\frac{7}{2}}dw" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s75EfJsBIlhx8qwjNzZ0nVAY5IDvasyveV76BLAM2+GK6y8y3IkIxawNlJGemMMxJLGvmg1GZ1E5pivwPmZC8H4kG8zH7g+RjqjHjUMB5xHuOgDtB1yNcRprXV5E9sdhQNVk065f/L6Fv7Bz27ISqpWpkS3dlcCKpQTQcheuut7MkAg4ur5mjpA9R2wQs9NJIVgU4Mqf3XXBJdd95DVJPkIA=" } }, { "type": "step", "result": "=\\int\\:w^{\\frac{7}{2}}dw" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:w^{\\frac{7}{2}}dw", "result": "=\\frac{2}{9}w^{\\frac{9}{2}}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{w^{\\frac{7}{2}+1}}{\\frac{7}{2}+1}" }, { "type": "interim", "title": "Simplify $$\\frac{w^{\\frac{7}{2}+1}}{\\frac{7}{2}+1}:{\\quad}\\frac{2}{9}w^{\\frac{9}{2}}$$", "input": "\\frac{w^{\\frac{7}{2}+1}}{\\frac{7}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{7}{2}+1:{\\quad}\\frac{9}{2}$$", "input": "\\frac{7}{2}+1", "result": "=\\frac{w^{\\frac{7}{2}+1}}{\\frac{9}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{7}{2}+\\frac{1\\cdot\\:2}{2}" }, { "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{7+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$7+1\\cdot\\:2=9$$", "input": "7+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=7+2" }, { "type": "step", "primary": "Add the numbers: $$7+2=9$$", "result": "=9" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7obkZX5Met1uk8zuhQLYStd6GQqufR6tr2vPxOUv7H+896Wcd/plV45tm+W5sEXkiZmdtzptgy12UZtKkBe8/D6D18RfRgeZOo3RfnVwQMKI=" } }, { "type": "step", "result": "=\\frac{9}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "$$w^{\\frac{7}{2}+1}=w^{\\frac{9}{2}}$$", "input": "w^{\\frac{7}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{7}{2}+1:{\\quad}\\frac{9}{2}$$", "input": "\\frac{7}{2}+1", "result": "=w^{\\frac{9}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{7}{2}+\\frac{1\\cdot\\:2}{2}" }, { "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{7+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$7+1\\cdot\\:2=9$$", "input": "7+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=7+2" }, { "type": "step", "primary": "Add the numbers: $$7+2=9$$", "result": "=9" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7obkZX5Met1uk8zuhQLYStd6GQqufR6tr2vPxOUv7H+896Wcd/plV45tm+W5sEXkiZmdtzptgy12UZtKkBe8/D6D18RfRgeZOo3RfnVwQMKI=" } }, { "type": "step", "result": "=\\frac{9}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gnhhdm1oKLneTc/8W+GfMSa+StGnWtengtTZdLN8SGHMwViaLUXkeD+JukROhWdjdM0rBJU68Tp8qcxi8Asia/8//6/nV5O4fb8Xgwi7maqy1ub1hTGJQy6bb9nB8fIiDp+jhbIugSbsecol3zYTd1iVI3uvN1by+AN9NfjoKFU=" } }, { "type": "step", "result": "=\\frac{w^{\\frac{9}{2}}}{\\frac{9}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{w^{\\frac{9}{2}}\\cdot\\:2}{9}" }, { "type": "step", "result": "=\\frac{2}{9}w^{\\frac{9}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{2}{9}w^{\\frac{9}{2}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7xEfVVgATqGs768hmVhkfV5aq0/H9iE3Xs3q/MQHBUhurrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx6JB7Asf8mi7IuDtv3AUYazMQgU64l06mXOZRUMRFGkoPmUz1GYa4snANpQYMxmVABSBv6izheLVUKQ/emokAUyVi4djWqB/aQf/oQfiXX55rCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "primary": "Substitute back $$w=\\ln\\left(v\\right)$$", "result": "=\\frac{2}{9}\\ln^{\\frac{9}{2}}\\left(v\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{\\ln^{3}\\left(v\\right)}{v}dv=\\frac{\\ln^{4}\\left(v\\right)}{4}$$", "input": "\\int\\:\\frac{\\ln^{3}\\left(v\\right)}{v}dv", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\ln^{3}\\left(v\\right)}{v}dv", "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: $$w=\\ln\\left(v\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{dw}{dv}=\\frac{1}{v}$$", "input": "\\frac{d}{dv}\\left(\\ln\\left(v\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dv}\\left(\\ln\\left(v\\right)\\right)=\\frac{1}{v}$$", "result": "=\\frac{1}{v}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqmzOQ63l6HV0QSyzcsb0SEcjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJitKzT0FSnQ1VzGfSlQmOzNTW26qciuyUBGXQExCUedYuGwmAwhCbJ/WP6zWvmSOUQgcJjF6wpuoCfd4QcipEuc=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dw=\\frac{1}{v}dv$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=vdw$$" }, { "type": "step", "result": "=\\int\\:\\frac{w^{3}}{v}vdw" }, { "type": "interim", "title": "Simplify $$\\frac{w^{3}}{v}v:{\\quad}w^{3}$$", "input": "\\frac{w^{3}}{v}v", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{w^{3}v}{v}" }, { "type": "step", "primary": "Cancel the common factor: $$v$$", "result": "=w^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:w^{3}dw" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s76aYbSPbE4TKfG1IN9RCmkJ9K2XLPCA6Y5+uUyKjYX2s+qPBBXW0OBa8HfjFPmx5EoAIezcf2HGXjCQp0SyS6as7d3vw7eocUrI0TgQGIqC/pkXGdJIGpBdOlYO6574DEdbA+zX4bD3u3gx65o2NJhOP8jPOjUAODCISjG3EPUUsKIcdyTU2qbTbjVtrM8zqhQ==" } }, { "type": "step", "result": "=\\int\\:w^{3}dw" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:w^{3}dw", "result": "=\\frac{w^{4}}{4}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{w^{3+1}}{3+1}" }, { "type": "interim", "title": "Simplify $$\\frac{w^{3+1}}{3+1}:{\\quad}\\frac{w^{4}}{4}$$", "input": "\\frac{w^{3+1}}{3+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=\\frac{w^{4}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{w^{4}}{4}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s78hkcOxlzQ1orA6wQV0uTMGo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ofLhcb65sUy0NoWNoNsV3VGgQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "step", "primary": "Substitute back $$w=\\ln\\left(v\\right)$$", "result": "=\\frac{\\ln^{4}\\left(v\\right)}{4}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=6v-\\frac{2}{9}\\ln^{\\frac{9}{2}}\\left(v\\right)-\\frac{\\ln^{4}\\left(v\\right)}{4}" }, { "type": "step", "primary": "Substitute back $$v=e^{u}$$", "result": "=6e^{u}-\\frac{2}{9}\\ln^{\\frac{9}{2}}\\left(e^{u}\\right)-\\frac{\\ln^{4}\\left(e^{u}\\right)}{4}" }, { "type": "interim", "title": "Simplify $$6e^{u}-\\frac{2}{9}\\ln^{\\frac{9}{2}}\\left(e^{u}\\right)-\\frac{\\ln^{4}\\left(e^{u}\\right)}{4}:{\\quad}6e^{u}-\\frac{2}{9}u^{\\frac{9}{2}}-\\frac{u^{4}}{4}$$", "input": "6e^{u}-\\frac{2}{9}\\ln^{\\frac{9}{2}}\\left(e^{u}\\right)-\\frac{\\ln^{4}\\left(e^{u}\\right)}{4}", "result": "=6e^{u}-\\frac{2}{9}u^{\\frac{9}{2}}-\\frac{u^{4}}{4}", "steps": [ { "type": "interim", "title": "$$\\ln^{\\frac{9}{2}}\\left(e^{u}\\right)=u^{\\frac{9}{2}}$$", "input": "\\ln^{\\frac{9}{2}}\\left(e^{u}\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\ln\\left(e^{u}\\right):{\\quad}u$$", "input": "\\ln\\left(e^{u}\\right)", "result": "=u^{\\frac{9}{2}}", "steps": [ { "type": "step", "primary": "Apply log rule $$\\log_{a}\\left(x^b\\right)=b\\cdot\\log_{a}\\left(x\\right),\\:\\quad$$ assuming $$x\\:\\geq\\:0$$", "result": "=\\ln\\left(e\\right)u" }, { "type": "step", "primary": "Apply log rule: $$\\log_a\\left(a\\right)=1$$", "result": "=u", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rq61fnk9+nGAJKQMHtXAkk/EK6umhJF9W5XDutNOY+8gJ/ZZA32ZInFBpDtxBfiK283O9WT8iUfLL0MzAdeeqPPfP47MVQWv5gcSYAj0xerGZW/OEdT0bPCk0BdA4Zj8eX8hW0vnpwjct1j+cKgg+SgGz8mZkqG8F9+NssS/NAlYlSN7rzdW8vgDfTX46ChV" } }, { "type": "step", "result": "=6e^{u}-\\frac{2}{9}u^{\\frac{9}{2}}-\\frac{\\ln^{4}\\left(e^{u}\\right)}{4}" }, { "type": "interim", "title": "$$\\ln^{4}\\left(e^{u}\\right)=u^{4}$$", "input": "\\ln^{4}\\left(e^{u}\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\ln\\left(e^{u}\\right):{\\quad}u$$", "input": "\\ln\\left(e^{u}\\right)", "result": "=u^{4}", "steps": [ { "type": "step", "primary": "Apply log rule $$\\log_{a}\\left(x^b\\right)=b\\cdot\\log_{a}\\left(x\\right),\\:\\quad$$ assuming $$x\\:\\geq\\:0$$", "result": "=\\ln\\left(e\\right)u" }, { "type": "step", "primary": "Apply log rule: $$\\log_a\\left(a\\right)=1$$", "result": "=u", "meta": { "practiceLink": "/practice/logarithms-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X2memFOPcrOJgaXUkpa0ONMvHyY50dhXPFfrjcmooUhjW2Y4F1QxfEC5t+Dx6+7F+yJwO2vX/hJkuWfCtbSyYz+z2VfW9oauX5yayLvpaxM3iSWTtuh2Hav4KeLLbHVd" } }, { "type": "step", "result": "=6e^{u}-\\frac{2}{9}u^{\\frac{9}{2}}-\\frac{u^{4}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CtST/wzIrnm/9YbQgLTkf1AC5460vT2YDlo3TfboIOBjvrh8JExcJGJXs9IQcQdmBHsKAsi+8Gr+V8QKmxd3UPH5wPhxtSelZqEgxy+qmqvNGoPE9TME3q+OPmgkv2RQrsdcNvPbe1bIHVH7AW2ZnP88mvSfwvDfEBv+zQ3+6gwVfVxT3Z3WBslays3PMUUGWLJGD4yj5LTalmGUDuyqMYEFMST8lDZxn1Yq5HMKVTvSANGjWsa8veXZR3wvsnVx8wZqgjGWFZdhuuZqllv2b4haqk40K7WRJXZ6ZYaaeHMNOmNXeWNLrZilOGlceA0FMsr8gXA5mmWT8+hXx1tfEjGoi7HuOeg/wrUDJsyk+1I=" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=6e^{u}-\\frac{2}{9}u^{\\frac{9}{2}}-\\frac{u^{4}}{4}+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": "u", "plotRequest": "y=6e^{u}-\\frac{2}{9}u^{\\frac{9}{2}}-\\frac{u^{4}}{4}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }