integral of 6x+3

{ "query": { "display": "$$\\int\\:6x+3dx$$", "symbolab_question": "BIG_OPERATOR#\\int 6x+3dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "3x^{2}+3x+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:6x+3dx=3x^{2}+3x+C$$", "input": "\\int\\:6x+3dx", "steps": [ { "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\\:6xdx+\\int\\:3dx" }, { "type": "interim", "title": "$$\\int\\:6xdx=3x^{2}$$", "input": "\\int\\:6xdx", "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": "=6\\cdot\\:\\int\\:xdx" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:xdx", "result": "=6\\cdot\\:\\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" } }, { "type": "interim", "title": "Simplify $$6\\cdot\\:\\frac{x^{2}}{2}:{\\quad}3x^{2}$$", "input": "6\\cdot\\:\\frac{x^{2}}{2}", "result": "=3x^{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x^{2}\\cdot\\:6}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{2}=3$$", "result": "=3x^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wHsyNILr7YqAxQ7kxdsiXpueiJEsgPE+iR4wbOw0E77NGoPE9TME3q+OPmgkv2RQxZ009Aruu1c6UQGs+TyooUUqTd96MWTKI6Kr2Ib0iQAi5KYlQO0vFE/Inns2SruqiwXuMqXjZvrzB5ZFHXtLztqArhIgWv0Fh5tkOCiqL/Q=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:3dx=3x$$", "input": "\\int\\:3dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=3x" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=3x^{2}+3x" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=3x^{2}+3x+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=3x^{2}+3x+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }