integral of x(x-1)^7

{ "query": { "display": "$$\\int\\:x\\left(x-1\\right)^{7}dx$$", "symbolab_question": "BIG_OPERATOR#\\int x(x-1)^{7}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{x^{9}}{9}-\\frac{7x^{8}}{8}+3x^{7}-\\frac{35x^{6}}{6}+7x^{5}-\\frac{21x^{4}}{4}+\\frac{7x^{3}}{3}-\\frac{x^{2}}{2}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:x\\left(x-1\\right)^{7}dx=\\frac{x^{9}}{9}-\\frac{7x^{8}}{8}+3x^{7}-\\frac{35x^{6}}{6}+7x^{5}-\\frac{21x^{4}}{4}+\\frac{7x^{3}}{3}-\\frac{x^{2}}{2}+C$$", "input": "\\int\\:x\\left(x-1\\right)^{7}dx", "steps": [ { "type": "interim", "title": "Expand $$x\\left(x-1\\right)^{7}:{\\quad}x^{8}-7x^{7}+21x^{6}-35x^{5}+35x^{4}-21x^{3}+7x^{2}-x$$", "input": "x\\left(x-1\\right)^{7}", "steps": [ { "type": "interim", "title": "$$\\left(x-1\\right)^{7}=x^{7}-7x^{6}+21x^{5}-35x^{4}+35x^{3}-21x^{2}+7x-1$$", "input": "\\left(x-1\\right)^{7}", "steps": [ { "type": "step", "primary": "Apply binomial theorem: $$\\left(a+b\\right)^{n}=\\sum_{i=0}^{n}\\binom{n}{i}a^{\\left(n-i\\right)}b^{i}$$", "secondary": [ "$$a=x,\\:\\:b=-1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Binomial%20Expansion", "practiceTopic": "Binomial Expansion" } }, { "type": "step", "result": "=\\sum_{i=0}^{7}\\binom{7}{i}x^{\\left(7-i\\right)}\\left(-1\\right)^{i}" }, { "type": "interim", "title": "Expand summation", "result": "=\\frac{7!}{0!\\left(7-0\\right)!}x^{7}\\left(-1\\right)^{0}+\\frac{7!}{1!\\left(7-1\\right)!}x^{6}\\left(-1\\right)^{1}+\\frac{7!}{2!\\left(7-2\\right)!}x^{5}\\left(-1\\right)^{2}+\\frac{7!}{3!\\left(7-3\\right)!}x^{4}\\left(-1\\right)^{3}+\\frac{7!}{4!\\left(7-4\\right)!}x^{3}\\left(-1\\right)^{4}+\\frac{7!}{5!\\left(7-5\\right)!}x^{2}\\left(-1\\right)^{5}+\\frac{7!}{6!\\left(7-6\\right)!}x^{1}\\left(-1\\right)^{6}+\\frac{7!}{7!\\left(7-7\\right)!}x^{0}\\left(-1\\right)^{7}", "steps": [ { "type": "step", "primary": "$$\\binom{n}{i}=\\frac{n!}{i!\\left(n-i\\right)!}$$" }, { "type": "step", "primary": "$$\\quad\\:i=0\\quad:\\quad\\frac{7!}{0!\\left(7-0\\right)!}x^{7}\\left(-1\\right)^{0}$$" }, { "type": "step", "primary": "$$\\quad\\:i=1\\quad:\\quad\\frac{7!}{1!\\left(7-1\\right)!}x^{6}\\left(-1\\right)^{1}$$" }, { "type": "step", "primary": "$$\\quad\\:i=2\\quad:\\quad\\frac{7!}{2!\\left(7-2\\right)!}x^{5}\\left(-1\\right)^{2}$$" }, { "type": "step", "primary": "$$\\quad\\:i=3\\quad:\\quad\\frac{7!}{3!\\left(7-3\\right)!}x^{4}\\left(-1\\right)^{3}$$" }, { "type": "step", "primary": "$$\\quad\\:i=4\\quad:\\quad\\frac{7!}{4!\\left(7-4\\right)!}x^{3}\\left(-1\\right)^{4}$$" }, { "type": "step", "primary": "$$\\quad\\:i=5\\quad:\\quad\\frac{7!}{5!\\left(7-5\\right)!}x^{2}\\left(-1\\right)^{5}$$" }, { "type": "step", "primary": "$$\\quad\\:i=6\\quad:\\quad\\frac{7!}{6!\\left(7-6\\right)!}x^{1}\\left(-1\\right)^{6}$$" }, { "type": "step", "primary": "$$\\quad\\:i=7\\quad:\\quad\\frac{7!}{7!\\left(7-7\\right)!}x^{0}\\left(-1\\right)^{7}$$" }, { "type": "step", "result": "=\\frac{7!}{0!\\left(7-0\\right)!}x^{7}\\left(-1\\right)^{0}+\\frac{7!}{1!\\left(7-1\\right)!}x^{6}\\left(-1\\right)^{1}+\\frac{7!}{2!\\left(7-2\\right)!}x^{5}\\left(-1\\right)^{2}+\\frac{7!}{3!\\left(7-3\\right)!}x^{4}\\left(-1\\right)^{3}+\\frac{7!}{4!\\left(7-4\\right)!}x^{3}\\left(-1\\right)^{4}+\\frac{7!}{5!\\left(7-5\\right)!}x^{2}\\left(-1\\right)^{5}+\\frac{7!}{6!\\left(7-6\\right)!}x^{1}\\left(-1\\right)^{6}+\\frac{7!}{7!\\left(7-7\\right)!}x^{0}\\left(-1\\right)^{7}" } ], "meta": { "interimType": "Expand Apply Binom 0Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{7!}{0!\\left(7-0\\right)!}x^{7}\\left(-1\\right)^{0}:{\\quad}x^{7}$$", "input": "\\frac{7!}{0!\\left(7-0\\right)!}x^{7}\\left(-1\\right)^{0}", "steps": [ { "type": "step", "primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$", "secondary": [ "$$\\left(-1\\right)^{0}=1$$" ], "result": "=1\\cdot\\:\\frac{7!}{0!\\left(7-0\\right)!}x^{7}" }, { "type": "interim", "title": "$$\\frac{7!}{0!\\left(7-0\\right)!}=1$$", "input": "\\frac{7!}{0!\\left(7-0\\right)!}", "steps": [ { "type": "interim", "title": "$$0!\\left(7-0\\right)!=7!$$", "input": "0!\\left(7-0\\right)!", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$7-0=7$$", "result": "=0!\\cdot\\:7!" }, { "type": "step", "primary": "Apply factorial rule: $$0!=1$$", "result": "=1\\cdot\\:7!" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:7!=7!$$", "result": "=7!" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7N5pCDRWqWcH7b3IYiHYVic0ag8T1MwTer44+aCS/ZFAgJh6NxEcaszPbX+vxmX660oRyQNcuR7lkz3BrtX8qC2cyyc3rD/Ca8DbQKQUi4/4=" } }, { "type": "step", "result": "=\\frac{7!}{7!}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7C18c4SkRZ9RxugCTqbrIUmIleSpRFM99vdg1NvrRUVCrju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nYRU+horcLOU1MDN16acrEN5hGQIN6Ld6KslODlW8fBB" } }, { "type": "step", "result": "=1\\cdot\\:1\\cdot\\:x^{7}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x^{7}\\cdot\\:1=x^{7}$$", "result": "=x^{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{7!}{1!\\left(7-1\\right)!}x^{6}\\left(-1\\right)^{1}:{\\quad}-7x^{6}$$", "input": "\\frac{7!}{1!\\left(7-1\\right)!}x^{6}\\left(-1\\right)^{1}", "steps": [ { "type": "step", "primary": "Apply rule $$a^{1}=a$$", "secondary": [ "$$\\left(-1\\right)^{1}=-1$$" ], "result": "=\\left(-1\\right)\\frac{7!}{1!\\left(7-1\\right)!}x^{6}" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{7!}{1!\\left(7-1\\right)!}x^{6}\\cdot\\:1" }, { "type": "interim", "title": "$$\\frac{7!}{1!\\left(7-1\\right)!}=\\frac{7}{1!}$$", "input": "\\frac{7!}{1!\\left(7-1\\right)!}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$7-1=6$$", "result": "=\\frac{7!}{1!\\cdot\\:6!}" }, { "type": "step", "primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$", "secondary": [ "$$\\frac{7!}{6!}=7$$" ], "result": "=\\frac{7}{1!}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7li+km9pMpW/Sp10kdz2C6M5duRkU1pdewN5jDCqkeCSrju+5Z51e/ZZSD3gRHwjB59izKiYC0WetIeiqVjugMD/L0MoYg+CUn6oyL3EO7YputprJgGEOPOnQ9ZB26VXLoseHra4P9xpXg/pologrCoV1O4tzqeY7aVRTflsUZHM=" } }, { "type": "step", "result": "=-1\\cdot\\:\\frac{7}{1!}x^{6}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-1\\cdot\\:\\frac{7x^{6}}{1!}" }, { "type": "interim", "title": "Simplify $$\\frac{7x^{6}}{1!}:{\\quad}7x^{6}$$", "input": "\\frac{7x^{6}}{1!}", "steps": [ { "type": "step", "primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$", "secondary": [ "$$1!=1$$" ], "result": "=\\frac{7x^{6}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=7x^{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-1\\cdot\\:7x^{6}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:7=7$$", "result": "=-7x^{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{7!}{2!\\left(7-2\\right)!}x^{5}\\left(-1\\right)^{2}:{\\quad}21x^{5}$$", "input": "\\frac{7!}{2!\\left(7-2\\right)!}x^{5}\\left(-1\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\frac{7!}{2!\\left(7-2\\right)!}=\\frac{42}{2!}$$", "input": "\\frac{7!}{2!\\left(7-2\\right)!}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$7-2=5$$", "result": "=\\frac{7!}{2!\\cdot\\:5!}" }, { "type": "step", "primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$", "secondary": [ "$$\\frac{7!}{5!}=7\\cdot\\:6$$" ], "result": "=\\frac{7\\cdot\\:6}{2!}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{42}{2!}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7km38OZIBtCzVEdexzQ0sPcszRjliy7+JYi1/RUEe25irju+5Z51e/ZZSD3gRHwjBFDzSqr0se9xluOjMw9c8xf8//6/nV5O4fb8Xgwi7mapGWz4tqROdknL52L1J8cFqsuXRv8kl1BX+gOH5khQ4hHh2ZKnWpxIymVLpM1P79gg=" } }, { "type": "step", "result": "=\\left(-1\\right)^{2}\\frac{42}{2!}x^{5}" }, { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "step", "result": "=1\\cdot\\:\\frac{42}{2!}x^{5}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=1\\cdot\\:\\frac{42x^{5}}{2!}" }, { "type": "interim", "title": "Simplify $$\\frac{42x^{5}}{2!}:{\\quad}21x^{5}$$", "input": "\\frac{42x^{5}}{2!}", "steps": [ { "type": "interim", "title": "$$2!=2$$", "input": "2!", "steps": [ { "type": "step", "primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$", "secondary": [ "$$2!=1\\cdot\\:2$$" ], "result": "=1\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7v6/eVh8vKMXnTnxuN3zFTMzBWJotReR4P4m6RE6FZ2MEnP7Xv4jqQdsgR6rCHLiZRJ9TLnjGcV7xuLm79Wde5Q==" } }, { "type": "step", "result": "=\\frac{42x^{5}}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{42}{2}=21$$", "result": "=21x^{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=1\\cdot\\:21x^{5}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:21=21$$", "result": "=21x^{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{7!}{3!\\left(7-3\\right)!}x^{4}\\left(-1\\right)^{3}:{\\quad}-35x^{4}$$", "input": "\\frac{7!}{3!\\left(7-3\\right)!}x^{4}\\left(-1\\right)^{3}", "steps": [ { "type": "interim", "title": "$$\\frac{7!}{3!\\left(7-3\\right)!}=\\frac{210}{3!}$$", "input": "\\frac{7!}{3!\\left(7-3\\right)!}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$7-3=4$$", "result": "=\\frac{7!}{3!\\cdot\\:4!}" }, { "type": "step", "primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$", "secondary": [ "$$\\frac{7!}{4!}=7\\cdot\\:6\\cdot\\:5$$" ], "result": "=\\frac{7\\cdot\\:6\\cdot\\:5}{3!}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{210}{3!}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mEYZjrbjYKwlR1xNrsZ+vF4YWFPIi4tDibC1gzbdfMarju+5Z51e/ZZSD3gRHwjBDbspc/dO/ZulD98dUOz3TQez7XpsChdff9dxDApMFsHAzpBYw0+tjguqgjKvDs598zSD10rnzGh+EkZlr28ovQwjd7RrOpUESWFgO/8cNWw=" } }, { "type": "step", "result": "=\\left(-1\\right)^{3}\\frac{210}{3!}x^{4}" }, { "type": "interim", "title": "$$\\left(-1\\right)^{3}=-1$$", "input": "\\left(-1\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=-a^{n},\\:$$if $$n$$ is odd", "secondary": [ "$$\\left(-1\\right)^{3}=-1^{3}$$" ], "result": "=-1^{3}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ClCeQdnpyQa/XVdLG7jeqM0ag8T1MwTer44+aCS/ZFDVPYg32V0aoZlEjRGoE3SC92wC37GgJ7iWYZErnwi6GR47EHm2iOOTglAC5fu7cyE=" } }, { "type": "step", "result": "=\\left(-1\\right)\\frac{210}{3!}x^{4}" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{210}{3!}x^{4}\\cdot\\:1" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-1\\cdot\\:\\frac{210x^{4}}{3!}" }, { "type": "interim", "title": "Simplify $$\\frac{210x^{4}}{3!}:{\\quad}35x^{4}$$", "input": "\\frac{210x^{4}}{3!}", "steps": [ { "type": "interim", "title": "$$3!=6$$", "input": "3!", "steps": [ { "type": "step", "primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$", "secondary": [ "$$3!=1\\cdot\\:2\\cdot\\:3$$" ], "result": "=1\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2\\cdot\\:3=6$$", "result": "=6" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RciCdt6M4n7wesJ3iBJPn8zBWJotReR4P4m6RE6FZ2No8psdWgO5IuIgJajcfvo9neavaRADROt2EOT6omsR8g==" } }, { "type": "step", "result": "=\\frac{210x^{4}}{6}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{210}{6}=35$$", "result": "=35x^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-1\\cdot\\:35x^{4}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:35=35$$", "result": "=-35x^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{7!}{4!\\left(7-4\\right)!}x^{3}\\left(-1\\right)^{4}:{\\quad}35x^{3}$$", "input": "\\frac{7!}{4!\\left(7-4\\right)!}x^{3}\\left(-1\\right)^{4}", "steps": [ { "type": "interim", "title": "$$\\frac{7!}{4!\\left(7-4\\right)!}=\\frac{210}{3!}$$", "input": "\\frac{7!}{4!\\left(7-4\\right)!}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$7-4=3$$", "result": "=\\frac{7!}{4!\\cdot\\:3!}" }, { "type": "step", "primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$", "secondary": [ "$$\\frac{7!}{4!}=7\\cdot\\:6\\cdot\\:5$$" ], "result": "=\\frac{7\\cdot\\:6\\cdot\\:5}{3!}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{210}{3!}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NUU+tRHYaAhCD6dXqFaok8xYzNk472pWwI7WTBOnkc+rju+5Z51e/ZZSD3gRHwjBDbspc/dO/ZulD98dUOz3TQez7XpsChdff9dxDApMFsH2P1j+U+uMIhv0diHZGoOoWnw10uP5BTc0iIw7d/fvwAwjd7RrOpUESWFgO/8cNWw=" } }, { "type": "step", "result": "=\\left(-1\\right)^{4}\\frac{210}{3!}x^{3}" }, { "type": "interim", "title": "$$\\left(-1\\right)^{4}=1$$", "input": "\\left(-1\\right)^{4}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{4}=1^{4}$$" ], "result": "=1^{4}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tK6HvHm+qjIqMUTuRg+bfM0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintPCfoGm3ioHGXTuH9heH7AyKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "step", "result": "=1\\cdot\\:\\frac{210}{3!}x^{3}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=1\\cdot\\:\\frac{210x^{3}}{3!}" }, { "type": "interim", "title": "Simplify $$\\frac{210x^{3}}{3!}:{\\quad}35x^{3}$$", "input": "\\frac{210x^{3}}{3!}", "steps": [ { "type": "interim", "title": "$$3!=6$$", "input": "3!", "steps": [ { "type": "step", "primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$", "secondary": [ "$$3!=1\\cdot\\:2\\cdot\\:3$$" ], "result": "=1\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2\\cdot\\:3=6$$", "result": "=6" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RciCdt6M4n7wesJ3iBJPn8zBWJotReR4P4m6RE6FZ2No8psdWgO5IuIgJajcfvo9neavaRADROt2EOT6omsR8g==" } }, { "type": "step", "result": "=\\frac{210x^{3}}{6}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{210}{6}=35$$", "result": "=35x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=1\\cdot\\:35x^{3}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:35=35$$", "result": "=35x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{7!}{5!\\left(7-5\\right)!}x^{2}\\left(-1\\right)^{5}:{\\quad}-21x^{2}$$", "input": "\\frac{7!}{5!\\left(7-5\\right)!}x^{2}\\left(-1\\right)^{5}", "steps": [ { "type": "interim", "title": "$$\\frac{7!}{5!\\left(7-5\\right)!}=\\frac{42}{2!}$$", "input": "\\frac{7!}{5!\\left(7-5\\right)!}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$7-5=2$$", "result": "=\\frac{7!}{5!\\cdot\\:2!}" }, { "type": "step", "primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$", "secondary": [ "$$\\frac{7!}{5!}=7\\cdot\\:6$$" ], "result": "=\\frac{7\\cdot\\:6}{2!}" }, { "type": "step", "primary": "Refine", "result": "=\\frac{42}{2!}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s705y2XyDaT/Ra68ecEmvM5EWmX4i4vgqOU/KHvWn5Miurju+5Z51e/ZZSD3gRHwjBFDzSqr0se9xluOjMw9c8xf8//6/nV5O4fb8Xgwi7marvVns6gt6cZLtoppn1/Nq7b1cjNe3NhSNeQwsX8tdJcXh2ZKnWpxIymVLpM1P79gg=" } }, { "type": "step", "result": "=\\left(-1\\right)^{5}\\frac{42}{2!}x^{2}" }, { "type": "interim", "title": "$$\\left(-1\\right)^{5}=-1$$", "input": "\\left(-1\\right)^{5}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=-a^{n},\\:$$if $$n$$ is odd", "secondary": [ "$$\\left(-1\\right)^{5}=-1^{5}$$" ], "result": "=-1^{5}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LQ+mpHkcdPi9bE7gVu2XnM0ag8T1MwTer44+aCS/ZFDVPYg32V0aoZlEjRGoE3SCom+2S9M43aFu4kR1XixWRB47EHm2iOOTglAC5fu7cyE=" } }, { "type": "step", "result": "=\\left(-1\\right)\\frac{42}{2!}x^{2}" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{42}{2!}x^{2}\\cdot\\:1" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-1\\cdot\\:\\frac{42x^{2}}{2!}" }, { "type": "interim", "title": "Simplify $$\\frac{42x^{2}}{2!}:{\\quad}21x^{2}$$", "input": "\\frac{42x^{2}}{2!}", "steps": [ { "type": "interim", "title": "$$2!=2$$", "input": "2!", "steps": [ { "type": "step", "primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$", "secondary": [ "$$2!=1\\cdot\\:2$$" ], "result": "=1\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7v6/eVh8vKMXnTnxuN3zFTMzBWJotReR4P4m6RE6FZ2MEnP7Xv4jqQdsgR6rCHLiZRJ9TLnjGcV7xuLm79Wde5Q==" } }, { "type": "step", "result": "=\\frac{42x^{2}}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{42}{2}=21$$", "result": "=21x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=-1\\cdot\\:21x^{2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:21=21$$", "result": "=-21x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{7!}{6!\\left(7-6\\right)!}x^{1}\\left(-1\\right)^{6}:{\\quad}7x$$", "input": "\\frac{7!}{6!\\left(7-6\\right)!}x^{1}\\left(-1\\right)^{6}", "steps": [ { "type": "step", "primary": "Apply rule $$a^{1}=a$$", "secondary": [ "$$x^{1}=x$$" ], "result": "=\\left(-1\\right)^{6}\\frac{7!}{6!\\left(7-6\\right)!}x" }, { "type": "interim", "title": "$$\\frac{7!}{6!\\left(7-6\\right)!}=\\frac{7}{1!}$$", "input": "\\frac{7!}{6!\\left(7-6\\right)!}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$7-6=1$$", "result": "=\\frac{7!}{6!\\cdot\\:1!}" }, { "type": "step", "primary": "Cancel the factorials: $$\\frac{n!}{\\left(n-m\\right)!}=n\\cdot\\left(n-1\\right)\\cdots\\left(n-m+1\\right),\\:n>m$$", "secondary": [ "$$\\frac{7!}{6!}=7$$" ], "result": "=\\frac{7}{1!}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oPDXaAQjcYAOjuGiVCehDmz2lrzC3ZVC0d7iusVb/Derju+5Z51e/ZZSD3gRHwjB59izKiYC0WetIeiqVjugMD/L0MoYg+CUn6oyL3EO7YqA73TMfh1JwYh68N4xgDZ8h8sSVODFFtxaKqUX/2uU0YV1O4tzqeY7aVRTflsUZHM=" } }, { "type": "step", "result": "=\\left(-1\\right)^{6}\\frac{7}{1!}x" }, { "type": "interim", "title": "$$\\left(-1\\right)^{6}=1$$", "input": "\\left(-1\\right)^{6}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{6}=1^{6}$$" ], "result": "=1^{6}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zwt2bVzcAflyZBi8z4JTrc0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintbfi7dPKgrS5Jm23O28N4qSKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "step", "result": "=1\\cdot\\:\\frac{7}{1!}x" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=1\\cdot\\:\\frac{7x}{1!}" }, { "type": "interim", "title": "Simplify $$\\frac{7x}{1!}:{\\quad}7x$$", "input": "\\frac{7x}{1!}", "steps": [ { "type": "step", "primary": "Apply factorial rule: $$n!=1\\cdot2\\cdot3\\cdot\\ldots\\cdot\\:n$$", "secondary": [ "$$1!=1$$" ], "result": "=\\frac{7x}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=7x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=1\\cdot\\:7x" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:7=7$$", "result": "=7x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "$$\\frac{7!}{7!\\left(7-7\\right)!}x^{0}\\left(-1\\right)^{7}=-1$$", "input": "\\frac{7!}{7!\\left(7-7\\right)!}x^{0}\\left(-1\\right)^{7}", "steps": [ { "type": "step", "primary": "Apply rule $$a^{0}=1,\\:a\\ne\\:0$$", "secondary": [ "$$x^{0}=1$$" ], "result": "=\\left(-1\\right)^{7}\\cdot\\:1\\cdot\\:\\frac{7!}{7!\\left(7-7\\right)!}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=1\\cdot\\:\\frac{\\left(-1\\right)^{7}\\cdot\\:7!}{7!\\left(7-7\\right)!}" }, { "type": "step", "primary": "Cancel the common factor: $$7!$$", "result": "=1\\cdot\\:\\frac{\\left(-1\\right)^{7}}{\\left(7-7\\right)!}" }, { "type": "interim", "title": "$$\\frac{\\left(-1\\right)^{7}}{\\left(7-7\\right)!}=-1$$", "input": "\\frac{\\left(-1\\right)^{7}}{\\left(7-7\\right)!}", "steps": [ { "type": "interim", "title": "$$\\left(-1\\right)^{7}=-1$$", "input": "\\left(-1\\right)^{7}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=-a^{n},\\:$$if $$n$$ is odd", "secondary": [ "$$\\left(-1\\right)^{7}=-1^{7}$$" ], "result": "=-1^{7}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Wo1LSdYhz13hU4JuCdjTVs0ag8T1MwTer44+aCS/ZFDVPYg32V0aoZlEjRGoE3SC2OiV9sCo2QnL4VbMCr971B47EHm2iOOTglAC5fu7cyE=" } }, { "type": "step", "result": "=\\frac{-1}{\\left(7-7\\right)!}" }, { "type": "interim", "title": "$$\\left(7-7\\right)!=1$$", "input": "\\left(7-7\\right)!", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$7-7=0$$", "result": "=0!" }, { "type": "step", "primary": "Apply factorial rule: $$0!=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DFmuXVLPbLqJMQ1Lf8PA6HWD310L1+P2yDQQfMEhENH7xkO7yIXgxJ7C3jKazWUhmZvyQRmh9XAXC4IY+vmcAyS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\frac{-1}{1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Q+cTk8Kr7HydPZwK5umR09D97V9+VHjLQDoVBqxn0EfNGoPE9TME3q+OPmgkv2RQ1T2IN9ldGqGZRI0RqBN0glNbbqpyK7JQEZdATEJR51jNeqegdwG8s0eI0TDEpk+pHjsQebaI45OCUALl+7tzIQ==" } }, { "type": "step", "result": "=1\\cdot\\:\\left(-1\\right)" }, { "type": "step", "primary": "Refine", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7en/hglOhtG+/IOE8jtHz4yiVpgZEfZSIUEudUZDC2SVSyqmWZetXEqFRFG11XuIQo5FYteSPKwXny4uCMrdsK+VM57iC4nRZKMRjJLY1Ha9azVVCkq5nLIe8kUmczNWEKJWmBkR9lIhQS51RkMLZJS5gBVAV48OqCqkYtt2T5fU=" } }, { "type": "step", "result": "=x^{7}-7x^{6}+21x^{5}-35x^{4}+35x^{3}-21x^{2}+7x-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76iZG14fl/Dt4XHrsmjf+nN13jtrSFDx+UNsawjlOjV1a8Tvodf84rOv+OZbL/eM0lu6NH5e2br5+EnyoF/l1568mu1F5rFglV59j3Zd4SzAcnY1IoDEjdTYo79X1naIMRtLCPRzaavbrJaerkdkOq01oOBz95Qh/GCmXZobhLYBHg+HD5RwrglW5Tjf/pene1QqO33+MAYk1hmZkA4jPROaqsReb2yyzr7lPaDshMWA=" } }, { "type": "step", "result": "=x\\left(x^{7}-7x^{6}+21x^{5}-35x^{4}+35x^{3}-21x^{2}+7x-1\\right)" }, { "type": "step", "primary": "Distribute parentheses", "result": "=xx^{7}+x\\left(-7x^{6}\\right)+x\\cdot\\:21x^{5}+x\\left(-35x^{4}\\right)+x\\cdot\\:35x^{3}+x\\left(-21x^{2}\\right)+x\\cdot\\:7x+x\\left(-1\\right)", "meta": { "title": { "extension": "Multiply each of the terms within the parentheses<br/>by the term outside the parenthesis" } } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$+\\left(-a\\right)=-a$$" ], "result": "=x^{7}x-7x^{6}x+21x^{5}x-35x^{4}x+35x^{3}x-21x^{2}x+7xx-1\\cdot\\:x" }, { "type": "interim", "title": "Simplify $$x^{7}x-7x^{6}x+21x^{5}x-35x^{4}x+35x^{3}x-21x^{2}x+7xx-1\\cdot\\:x:{\\quad}x^{8}-7x^{7}+21x^{6}-35x^{5}+35x^{4}-21x^{3}+7x^{2}-x$$", "input": "x^{7}x-7x^{6}x+21x^{5}x-35x^{4}x+35x^{3}x-21x^{2}x+7xx-1\\cdot\\:x", "result": "=x^{8}-7x^{7}+21x^{6}-35x^{5}+35x^{4}-21x^{3}+7x^{2}-x", "steps": [ { "type": "interim", "title": "$$x^{7}x=x^{8}$$", "input": "x^{7}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{7}x=\\:x^{7+1}$$" ], "result": "=x^{7+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$7+1=8$$", "result": "=x^{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IIFMUjR26glQCiLyKBUQQXWD310L1+P2yDQQfMEhENG/cwjVROhdZggWS3R5tmG6bFqwRFoX55vYSqiPinZ7GkVW+tOTJZuns8Oyk/1p7xY=" } }, { "type": "interim", "title": "$$7x^{6}x=7x^{7}$$", "input": "7x^{6}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{6}x=\\:x^{6+1}$$" ], "result": "=7x^{6+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$6+1=7$$", "result": "=7x^{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/WIwHPhG40d9ClFVWohEIQOfOVs9mPIqDLV5QIWwt3nxD49GkVEPfP9LUBxFYrH8FiFbBNk9gw83WH3uiNzOAMEJ36HhGUeHgViuGAzLK7I=" } }, { "type": "interim", "title": "$$21x^{5}x=21x^{6}$$", "input": "21x^{5}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{5}x=\\:x^{5+1}$$" ], "result": "=21x^{5+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=21x^{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Lad8rl8oX81V9TFgaLSVs0ag8T1MwTer44+aCS/ZFDd+NdIcDHR16ZNQ34szJw3Ks2VuLSL/nnquvYpJ7ag0zzktr09G7v4ahk6+i4vZIUkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$35x^{4}x=35x^{5}$$", "input": "35x^{4}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{4}x=\\:x^{4+1}$$" ], "result": "=35x^{4+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$4+1=5$$", "result": "=35x^{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73gFpixq9Vym+/AYwPPKKvs0ag8T1MwTer44+aCS/ZFDFhFi4aTUAiO6xJKQRWa/BYvigcTOfm92z9AM6ISd6VCNx9QwfuflD4WGYfp4D5fckt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$35x^{3}x=35x^{4}$$", "input": "35x^{3}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{3}x=\\:x^{3+1}$$" ], "result": "=35x^{3+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=35x^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pNj+hlb8eg3w2cHVBrP6O80ag8T1MwTer44+aCS/ZFD9VF4mulwq6tRVt/CqXJ10YvigcTOfm92z9AM6ISd6VEjShZWB9hWGVwTvKIvCJrskt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$21x^{2}x=21x^{3}$$", "input": "21x^{2}x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$x^{2}x=\\:x^{2+1}$$" ], "result": "=21x^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=21x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JzLQ2YW9VbeWaeL4A+cqtM0ag8T1MwTer44+aCS/ZFCHcRC1OnZnfGXwzPecCNF1Ks2VuLSL/nnquvYpJ7ag06zew055EeUSJlXWCuqo/n4kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$7xx=7x^{2}$$", "input": "7xx", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$xx=\\:x^{1+1}$$" ], "result": "=7x^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=7x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qBN7KimH17HHB3t62/dSTHCQoYlYQ8U+Tfyx0kyzI8jLLR0C3oWdZTQxqSN6Hsehr8mMxtr3fC25QUJ3OhFyX48BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "interim", "title": "$$1\\cdot\\:x=x$$", "input": "1\\cdot\\:x", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ASx2YODBupsHY/9yrO15bd13jtrSFDx+UNsawjlOjV3pfPCe8nQAZY1bE89UDVgMPJrYhwc+zvuHrOLz58Ml2oD661lPR3w/W4zyCV9dwUw=" } }, { "type": "step", "result": "=x^{8}-7x^{7}+21x^{6}-35x^{5}+35x^{4}-21x^{3}+7x^{2}-x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tjw3HjsoesIdYpnXnuxqGSAn9lkDfZkicUGkO3EF+IpP3rh7ROWwjslzxEO7VkWgmdY0oBWPKJX3ib+7lqYA0HMIXlwVLB8R0oTdkW02IbRbkgsK1YcZrwQf2vbiflRso3oe/oyhMy2+1TQhDBd2f/7hpPfQ0As+kA1eQGg+EzSbd55aYT0LXMKoTr5Or8y5" } }, { "type": "step", "result": "=\\int\\:x^{8}-7x^{7}+21x^{6}-35x^{5}+35x^{4}-21x^{3}+7x^{2}-xdx" }, { "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\\:x^{8}dx-\\int\\:7x^{7}dx+\\int\\:21x^{6}dx-\\int\\:35x^{5}dx+\\int\\:35x^{4}dx-\\int\\:21x^{3}dx+\\int\\:7x^{2}dx-\\int\\:xdx" }, { "type": "interim", "title": "$$\\int\\:x^{8}dx=\\frac{x^{9}}{9}$$", "input": "\\int\\:x^{8}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{8}dx", "result": "=\\frac{x^{9}}{9}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{8+1}}{8+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{8+1}}{8+1}:{\\quad}\\frac{x^{9}}{9}$$", "input": "\\frac{x^{8+1}}{8+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$8+1=9$$", "result": "=\\frac{x^{9}}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{9}}{9}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73cCNfa+e7U9kCDWuCNQ3mCo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ofR8igfcQRDrSpadkUMRABQgQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:7x^{7}dx=\\frac{7x^{8}}{8}$$", "input": "\\int\\:7x^{7}dx", "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": "=7\\cdot\\:\\int\\:x^{7}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{7}dx", "result": "=7\\cdot\\:\\frac{x^{8}}{8}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{7+1}}{7+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{7+1}}{7+1}:{\\quad}\\frac{x^{8}}{8}$$", "input": "\\frac{x^{7+1}}{7+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$7+1=8$$", "result": "=\\frac{x^{8}}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{8}}{8}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s78/zM1Ulm/GZYO5rUZkTS+qo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7offE7fULzLKTed+7Kn4hc0RgQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{7x^{8}}{8}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:21x^{6}dx=3x^{7}$$", "input": "\\int\\:21x^{6}dx", "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": "=21\\cdot\\:\\int\\:x^{6}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{6}dx", "result": "=21\\cdot\\:\\frac{x^{7}}{7}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{6+1}}{6+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{6+1}}{6+1}:{\\quad}\\frac{x^{7}}{7}$$", "input": "\\frac{x^{6+1}}{6+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$6+1=7$$", "result": "=\\frac{x^{7}}{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{7}}{7}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s71PX4w684PGoRK55u2wyT0mo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ocYj7vVG0kubj3kttWOlD7CgQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "interim", "title": "Simplify $$21\\cdot\\:\\frac{x^{7}}{7}:{\\quad}3x^{7}$$", "input": "21\\cdot\\:\\frac{x^{7}}{7}", "result": "=3x^{7}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x^{7}\\cdot\\:21}{7}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{21}{7}=3$$", "result": "=3x^{7}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mxgGsQ/39FKyrvDEgUxOP+UuIL2OUhLozZ0BjZBYo0bdd47a0hQ8flDbGsI5To1dBYBQA5F9DR7EXp0p9BXCcaN6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwgDLsqa9nL8hOCWIEgBNH6bSdOT91mb9zJdSjuQr4WVkw=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:35x^{5}dx=\\frac{35x^{6}}{6}$$", "input": "\\int\\:35x^{5}dx", "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": "=35\\cdot\\:\\int\\:x^{5}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{5}dx", "result": "=35\\cdot\\:\\frac{x^{6}}{6}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{5+1}}{5+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{5+1}}{5+1}:{\\quad}\\frac{x^{6}}{6}$$", "input": "\\frac{x^{5+1}}{5+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$5+1=6$$", "result": "=\\frac{x^{6}}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{6}}{6}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s70HUskMn2V8ZBlYunFrfJf6o/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ocymAVtGK/9bBXqLD28AmPngQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{35x^{6}}{6}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:35x^{4}dx=7x^{5}$$", "input": "\\int\\:35x^{4}dx", "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": "=35\\cdot\\:\\int\\:x^{4}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{4}dx", "result": "=35\\cdot\\:\\frac{x^{5}}{5}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{x^{4+1}}{4+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{4+1}}{4+1}:{\\quad}\\frac{x^{5}}{5}$$", "input": "\\frac{x^{4+1}}{4+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$4+1=5$$", "result": "=\\frac{x^{5}}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{5}}{5}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s792bwOWB9XLV4lUXfjhDNcao/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7oesKQBqfZltZgKEVL7oPcXqgQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "interim", "title": "Simplify $$35\\cdot\\:\\frac{x^{5}}{5}:{\\quad}7x^{5}$$", "input": "35\\cdot\\:\\frac{x^{5}}{5}", "result": "=7x^{5}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x^{5}\\cdot\\:35}{5}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{35}{5}=7$$", "result": "=7x^{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Tevw8Jprc6oU5d+QuY/NfyY8FK2So5gMF5bkV+8B4+fdd47a0hQ8flDbGsI5To1dhBNeJaAeFxg86jkzzMS5D6N6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwgGlX4uhLpYM/yWm496tMQ8FU7rtb3HBo1SaHqUYcM7qs=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:21x^{3}dx=\\frac{21x^{4}}{4}$$", "input": "\\int\\:21x^{3}dx", "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": "=21\\cdot\\:\\int\\:x^{3}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{3}dx", "result": "=21\\cdot\\:\\frac{x^{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{x^{3+1}}{3+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{3+1}}{3+1}:{\\quad}\\frac{x^{4}}{4}$$", "input": "\\frac{x^{3+1}}{3+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=\\frac{x^{4}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{4}}{4}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7xN+VLtRNPamYpzmOv38kp6o/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ofVrcspqtVis03j3TOAGHEUgQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{21x^{4}}{4}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:7x^{2}dx=\\frac{7x^{3}}{3}$$", "input": "\\int\\:7x^{2}dx", "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": "=7\\cdot\\:\\int\\:x^{2}dx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:x^{2}dx", "result": "=7\\cdot\\:\\frac{x^{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{x^{2+1}}{2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{2+1}}{2+1}:{\\quad}\\frac{x^{3}}{3}$$", "input": "\\frac{x^{2+1}}{2+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=\\frac{x^{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{3}}{3}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+w+ikB2VyJnNfLrQuoxvVyo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7odVISTIak7VD9OG2tlObqsigQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{7x^{3}}{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:xdx=\\frac{x^{2}}{2}$$", "input": "\\int\\:xdx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:xdx", "result": "=\\frac{x^{2}}{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{x^{1+1}}{1+1}" }, { "type": "interim", "title": "Simplify $$\\frac{x^{1+1}}{1+1}:{\\quad}\\frac{x^{2}}{2}$$", "input": "\\frac{x^{1+1}}{1+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{x^{2}}{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7814/6/Jz6acDoAMznrJ9GL/JyKXuO90NgYuEtRnVFUoQEgTxsQDcbkC7lns/WqbpPzIcDl+e6/8g9uDsiVdOq//YrZ1UCh4L70vx5eDNyDLTeQKHeh69S6dnv9vSoUoFEMybLZHp2MhZ1cw+jOu7RuDCZKz/+DESbePVmsYY2Aq" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{x^{9}}{9}-\\frac{7x^{8}}{8}+3x^{7}-\\frac{35x^{6}}{6}+7x^{5}-\\frac{21x^{4}}{4}+\\frac{7x^{3}}{3}-\\frac{x^{2}}{2}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{x^{9}}{9}-\\frac{7x^{8}}{8}+3x^{7}-\\frac{35x^{6}}{6}+7x^{5}-\\frac{21x^{4}}{4}+\\frac{7x^{3}}{3}-\\frac{x^{2}}{2}+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=Sum%20Rule", "practiceTopic": "Integral Sum Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{x^{9}}{9}-\\frac{7x^{8}}{8}+3x^{7}-\\frac{35x^{6}}{6}+7x^{5}-\\frac{21x^{4}}{4}+\\frac{7x^{3}}{3}-\\frac{x^{2}}{2}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }