integral of e^{2x}

{ "query": { "display": "integral $$e^{2x}$$", "symbolab_question": "PRE_CALC#integral e^{2x}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{1}{2}e^{2x}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:e^{2x}dx=\\frac{1}{2}e^{2x}+C$$", "input": "\\int\\:e^{2x}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:e^{2x}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=2x$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51iZ4v02Fm2dNqQJZnxCX4Je" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{2}du$$" }, { "type": "step", "result": "=\\int\\:e^{u}\\frac{1}{2}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7/QtD4Yq7g9dMrr8+Y+WKeH6o8EFdbQ4Frwd+MU+bHkSgAh7Nx/YcZeMJCnRLJLpqzt3e/Dt6hxSsjROBAYioL85S+GyYIKkVwlHzDAfDHqTRnTpbscABm7igdDhd7igC+bDWr6GMUSMrMlGBgtNEZJP943lRnGl60Ia9GIhzzMJcyQyKTTPwYipg/6j/ntaLQ==" } }, { "type": "step", "result": "=\\int\\:e^{u}\\frac{1}{2}du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=\\frac{1}{2}\\cdot\\:\\int\\:e^{u}du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:e^{u}du=e^{u}$$", "result": "=\\frac{1}{2}e^{u}" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=\\frac{1}{2}e^{2x}" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{1}{2}e^{2x}+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", "practiceTopic": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{1}{2}e^{2x}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }