integral from 0 to e of a^x

{ "query": { "display": "$$\\int_{0}^{e}a^{x}dx$$", "symbolab_question": "BIG_OPERATOR#\\int _{0}^{e}a^{x}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Definite Integrals", "default": "\\frac{a^{e}-1}{\\ln(a)}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int_{0}^{e}a^{x}dx=\\frac{a^{e}-1}{\\ln\\left(a\\right)}$$", "input": "\\int_{0}^{e}a^{x}dx", "steps": [ { "type": "step", "primary": "$$\\int{a^x}dx=\\frac{a^x}{\\ln{a}}$$", "result": "=[\\frac{a^{x}}{\\ln\\left(a\\right)}]_{0}^{e}", "meta": { "general_rule": { "extension": "$$\\int{a^x}dx=\\int{e^{\\ln{a^x}}}=\\int{e^{x\\ln{a}}}=\\frac{e^{x\\ln{a}}}{\\ln{a}}=\\frac{e^{\\ln{a^x}}}{\\ln{a}}=\\frac{a^x}{\\ln{a}}$$" } } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\frac{a^{e}}{\\ln\\left(a\\right)}-\\frac{1}{\\ln\\left(a\\right)}$$", "input": "[\\frac{a^{x}}{\\ln\\left(a\\right)}]_{0}^{e}", "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": "$$\\lim_{x\\to\\:0+}\\left(\\frac{a^{x}}{\\ln\\left(a\\right)}\\right)=\\frac{1}{\\ln\\left(a\\right)}$$", "input": "\\lim_{x\\to\\:0+}\\left(\\frac{a^{x}}{\\ln\\left(a\\right)}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=0$$", "result": "=\\frac{a^{0}}{\\ln\\left(a\\right)}", "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}{\\ln\\left(a\\right)}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{x\\to\\:e-}\\left(\\frac{a^{x}}{\\ln\\left(a\\right)}\\right)=\\frac{a^{e}}{\\ln\\left(a\\right)}$$", "input": "\\lim_{x\\to\\:e-}\\left(\\frac{a^{x}}{\\ln\\left(a\\right)}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$x=e$$", "result": "=\\frac{a^{e}}{\\ln\\left(a\\right)}", "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 $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\frac{a^{e}}{\\ln\\left(a\\right)}-\\frac{1}{\\ln\\left(a\\right)}" } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7xTmJPb0/sgLMeTwoEloNBQm9V/Bk1ibjaeyWJhgrw41uQ4kmq/soqHzBZwEaFJcUmEKsnCWXjtKIxhF7fKNxWaoEv4hTSrtIpYubLu7PWpDgQUxJPyUNnGfVirkcwpVO4F9y1KFIeHqjUpV4rro83BWfqQVURCgodkjT0tzNhTj" } }, { "type": "step", "result": "=\\frac{a^{e}}{\\ln\\left(a\\right)}-\\frac{1}{\\ln\\left(a\\right)}" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{a^{e}-1}{\\ln\\left(a\\right)}" } ], "meta": { "solvingClass": "Integrals" } }, "meta": { "showVerify": true } }