integral from 0 to 1 of x(x^2+1)

{ "query": { "display": "$$\\int_{0}^{1}x\\left(x^{2}+1\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int _{0}^{1}x(x^{2}+1)dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Definite Integrals", "default": "\\frac{3}{4}", "decimal": "0.75", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int_{0}^{1}x\\left(x^{2}+1\\right)dx=\\frac{3}{4}$$", "input": "\\int_{0}^{1}x\\left(x^{2}+1\\right)dx", "steps": [ { "type": "interim", "title": "Expand $$x\\left(x^{2}+1\\right):{\\quad}x^{3}+x$$", "input": "x\\left(x^{2}+1\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=x,\\:b=x^{2},\\:c=1$$" ], "result": "=xx^{2}+x\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=x^{2}x+1\\cdot\\:x" }, { "type": "interim", "title": "Simplify $$x^{2}x+1\\cdot\\:x:{\\quad}x^{3}+x$$", "input": "x^{2}x+1\\cdot\\:x", "result": "=x^{3}+x", "steps": [ { "type": "interim", "title": "$$x^{2}x=x^{3}$$", "input": "x^{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": "=x^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=x^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LWrIAcT/Pu2l3a2UlrsCC3WD310L1+P2yDQQfMEhENGDPbrpZaMLRegCZc+JnvJI5kmDAAHjIPJcICsCIhoRbVuSVZd9z4+kRKtqsjU2P18=" } }, { "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^{3}+x" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7V+PxElJnnpEAfDxDaeL3/yAn9lkDfZkicUGkO3EF+Ipo8uWRXa/vCvc6+3V1g9fiP8vQyhiD4JSfqjIvcQ7tiqvhme4+pJbvIQ3GyaucLUG3HSKi85UsW/hUAI5WNk4q" } }, { "type": "step", "result": "=\\int_{0}^{1}x^{3}+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_{0}^{1}x^{3}dx+\\int_{0}^{1}xdx" }, { "type": "interim", "title": "$$\\int_{0}^{1}x^{3}dx=\\frac{1}{4}$$", "input": "\\int_{0}^{1}x^{3}dx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{0}^{1}x^{3}dx", "result": "=[\\frac{x^{4}}{4}]_{0}^{1}", "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}]_{0}^{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}]_{0}^{1}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s714ZdI58A6px+C7dE/wSh3mTdaV09PMxEKZ9FieghTFwnz9JShbQBLExa7JLGJEbAosjLe8tD9HbrkG8vq6q9jjtfmb9XXfd5ttvMBzR/l4YZxJnQY1+wRgPIKU6QDpxvEUqTd96MWTKI6Kr2Ib0iQBbgTZHtHmQDZkDHT6lbFNTOZoJMX7Twq11mSDHSln+YA==" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\frac{1}{4}$$", "input": "[\\frac{x^{4}}{4}]_{0}^{1}", "steps": [ { "type": "step", "primary": "$$\\int_{a}^{b}{f\\left(x\\right)dx}=F\\left(b\\right)-F\\left(a\\right)=\\lim_{x\\to\\:b-}\\left(F\\left(x\\right)\\right)-\\lim_{x\\to\\:a+}\\left(F\\left(x\\right)\\right)$$" }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{x^{4}}{4}\\right)=0$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{x^{4}}{4}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=\\frac{0^{4}}{4}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{0^{4}}{4}:{\\quad}0$$", "input": "\\frac{0^{4}}{4}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{4}=0$$" ], "result": "=\\frac{0}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SIJzsSdJI+WyHot1aBR+lGFExV4mFNoq+OGxoJmhOAkE5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXRGCE6eBJR17v41x11J2XrZzpWlbt/AlNC3EwOtC8OHxw==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:1-}\\left(\\frac{x^{4}}{4}\\right)=\\frac{1}{4}$$", "input": "\\lim_{x\\to\\:1-}\\left(\\frac{x^{4}}{4}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=1$$", "result": "=\\frac{1^{4}}{4}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{4}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{1}{4}-0" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{4}", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s714ZdI58A6px+C7dE/wSh3km9V/Bk1ibjaeyWJhgrw41uQ4kmq/soqHzBZwEaFJcUhrDTchWYTtEunqHYbO9i/2gtm0MPhUSpjQTUqAbvqvJi4n0v56P5SoFAesqLAyUYwanIPMU//UDMHbnrRFJUfzqvuVTzkV43woA0R3vH6y9" } }, { "type": "step", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int_{0}^{1}xdx=\\frac{1}{2}$$", "input": "\\int_{0}^{1}xdx", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{0}^{1}xdx", "result": "=[\\frac{x^{2}}{2}]_{0}^{1}", "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}]_{0}^{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}]_{0}^{1}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+Q+W0W2FArTKy+svCrd1QMsjvX7KVUO/AeCFSId4S33HipIftvBYl8MvlbM/MS0IniX35dQ/h01lIvxamZtt5P0OFtj1jyjvPrkcLqztccOf5UEZ2/O3gFmSOyfIvXNqdbA+zX4bD3u3gx65o2NJhN+u9S1UanyCrDStYeLlcNDOFynCe2Jk2u4EAkbH+yVgg==" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\frac{1}{2}$$", "input": "[\\frac{x^{2}}{2}]_{0}^{1}", "steps": [ { "type": "step", "primary": "$$\\int_{a}^{b}{f\\left(x\\right)dx}=F\\left(b\\right)-F\\left(a\\right)=\\lim_{x\\to\\:b-}\\left(F\\left(x\\right)\\right)-\\lim_{x\\to\\:a+}\\left(F\\left(x\\right)\\right)$$" }, { "type": "interim", "title": "$$\\lim_{x\\to\\:0+}\\left(\\frac{x^{2}}{2}\\right)=0$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{x^{2}}{2}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=\\frac{0^{2}}{2}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{0^{2}}{2}:{\\quad}0$$", "input": "\\frac{0^{2}}{2}", "result": "=0", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{2}=0$$" ], "result": "=\\frac{0}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nteUyMJWMp4mm/uXtkFpCVnyYRz18HvB+rp63mPitc8E5aqGN/sLZfeoFZRwtGLqP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXR3DTZYr1PJ9/OYrvIJiwrr+i0Ux3lprvX50CFfl5rrAQ==" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:1-}\\left(\\frac{x^{2}}{2}\\right)=\\frac{1}{2}$$", "input": "\\lim_{x\\to\\:1-}\\left(\\frac{x^{2}}{2}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=1$$", "result": "=\\frac{1^{2}}{2}", "meta": { "title": { "extension": "Limit properties - if the limit of f(x), and g(x) exists, then:<br/>$$\\bullet\\quad\\lim_{x\\to\\:a}\\left(x\\right)=a$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[\\left(f\\left(x\\right)\\right)^c]=\\left(\\lim_{x\\to{a}}{f\\left(x\\right)}\\right)^c$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}[f\\left(x\\right)\\cdot{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\cdot\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>$$\\bullet\\quad\\lim_{x\\to{a}}\\left(\\frac{f\\left(x\\right)}{g\\left(x\\right)}\\right)=\\frac{\\lim_{x\\to{a}}{f\\left(x\\right)}}{\\lim_{x\\to{a}}{g\\left(x\\right)}},\\:$$where $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{1}{2}-0" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+Q+W0W2FArTKy+svCrd1QOPVo8rHFjKlUhEt/c08PpWH/+9ZCmtxJEW2yM0lAeSa4tu9GwZ3vkMP5rfw6uavIH/P/+v51eTuH2/F4MIu5mqU/td0/rq+faUMiA6Cy3iZF1qkcD3DkFSvldxTLmJepCJqVxX90jlMfh9fKn6dzC4" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=\\frac{1}{4}+\\frac{1}{2}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{4}+\\frac{1}{2}:{\\quad}\\frac{3}{4}$$", "input": "\\frac{1}{4}+\\frac{1}{2}", "result": "=\\frac{3}{4}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$4,\\:2:{\\quad}4$$", "input": "4,\\:2", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$4:{\\quad}2\\cdot\\:2$$", "input": "4", "steps": [ { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp39fF/zAtU5baHQ1hwgXA+n" } }, { "type": "interim", "title": "Prime factorization of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFw/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1+G9v2aKasChgV65VW8cTW" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$4$$ or $$2$$", "result": "=2\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$4$$" }, { "type": "step", "primary": "For $$\\frac{1}{2}:\\:$$multiply the denominator and numerator by $$2$$", "result": "\\frac{1}{2}=\\frac{1\\cdot\\:2}{2\\cdot\\:2}=\\frac{2}{4}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{1}{4}+\\frac{2}{4}" }, { "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{1+2}{4}" }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=\\frac{3}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72S64wvM73kdcljoY0cQeaJZNycihl64L0wizN6VZBHzdd47a0hQ8flDbGsI5To1drfMRPynLSfP3VYe1OQldRFO1T0snFOPqKXL+S6MxVmmY3ASC+aZqPN1DBWUUsybFLOoCuoLwpnU8v8HDSwilTDdNgIIXGOPwKpwlvAzQVNewiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }