integral from-1 to 3 of x/(x^2+4)

{ "query": { "display": "$$\\int_{-1}^{3}\\frac{x}{x^{2}+4}dx$$", "symbolab_question": "BIG_OPERATOR#\\int _{-1}^{3}\\frac{x}{x^{2}+4}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Definite Integrals", "default": "\\frac{1}{2}(\\ln(13)-\\ln(5))", "decimal": "0.47775…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int_{-1}^{3}\\frac{x}{x^{2}+4}dx=\\frac{1}{2}\\left(\\ln\\left(13\\right)-\\ln\\left(5\\right)\\right)$$", "input": "\\int_{-1}^{3}\\frac{x}{x^{2}+4}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int_{-1}^{3}\\frac{x}{x^{2}+4}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: $$u=x^{2}+4$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+4\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)+\\frac{d}{dx}\\left(4\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4\\right)=0$$", "input": "\\frac{d}{dx}\\left(4\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjVwwDW+HeFUFiKZ8J+l8XpJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTt4/nDM7CraQVY2V0O4nKcI" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2xdx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{2x}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{x}{u}\\cdot\\:\\frac{1}{2x}du" }, { "type": "interim", "title": "Simplify $$\\frac{x}{u}\\cdot\\:\\frac{1}{2x}:{\\quad}\\frac{1}{2u}$$", "input": "\\frac{x}{u}\\cdot\\:\\frac{1}{2x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{x\\cdot\\:1}{u\\cdot\\:2x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{1}{u\\cdot\\:2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{2u}du" }, { "type": "step", "primary": "Adjust integral boundaries:" }, { "type": "interim", "title": "$$x=-1\\quad\\Rightarrow\\:u=5$$", "input": "u=x^{2}+4", "steps": [ { "type": "step", "primary": "Plug in $$x=-1$$", "result": "=\\left(-1\\right)^{2}+4" }, { "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+4" }, { "type": "step", "primary": "Add the numbers: $$1+4=5$$", "result": "=5" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VWm9FKKy9YwyFso5V7cHYt13jtrSFDx+UNsawjlOjV11nC4bDH55BwQG/51jcKNME0RqYrki7jZud8eTFGbF1lUUew5fVKmjQ3z3Sfxbm0DPvCqOAn9E9e9YlRaXv0KE" } }, { "type": "interim", "title": "$$x=3\\quad\\Rightarrow\\:u=13$$", "input": "u=x^{2}+4", "steps": [ { "type": "step", "primary": "Plug in $$x=3$$", "result": "=3^{2}+4" }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=9+4" }, { "type": "step", "primary": "Add the numbers: $$9+4=13$$", "result": "=13" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VWm9FKKy9YwyFso5V7cHYt13jtrSFDx+UNsawjlOjV0DAV5nuz1c9ZX7Q9Xovy9RPqHxTNtxBriQCNRN7AHP6FUUew5fVKmjQ3z3Sfxbm0BbS97bkONcvQt0ANxToyyk" } }, { "type": "step", "result": "=\\int_{5}^{13}\\frac{1}{2u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s79cFAI8gYl5HxBYl8KTTJuXfQ5auaQyfm2QxyWXiiCUASUM9pakkKILvT6Fs/PM35zdOFr14NhlnTuLeZXt0DnkvTqyge4+vpHW4VBaWAsV8yoWOgcG1qRd/3+4F8wXfM95BL6c/JKNQuyvkBctlpqxkS3dlcCKpQTQcheuut7MkAg4ur5mjpA9R2wQs9NJIVgU4Mqf3XXBJdd95DVJPkIA=" } }, { "type": "step", "result": "=\\int_{5}^{13}\\frac{1}{2u}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_{5}^{13}\\frac{1}{u}du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u}du=\\ln\\left(\\left|u\\right|\\right)$$", "result": "=\\frac{1}{2}[\\ln\\left|u\\right|]_{5}^{13}" }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\ln\\left(13\\right)-\\ln\\left(5\\right)$$", "input": "[\\ln\\left|u\\right|]_{5}^{13}", "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": "$$\\lim_{u\\to\\:5+}\\left(\\ln\\left|u\\right|\\right)=\\ln\\left(5\\right)$$", "input": "\\lim_{u\\to\\:5+}\\left(\\ln\\left|u\\right|\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=5$$", "result": "=\\ln\\left|5\\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": "Apply absolute rule: $$\\left|a\\right|=a,\\:a\\ge0$$", "secondary": [ "$$\\left|5\\right|=5$$" ], "result": "=\\ln\\left(5\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{u\\to\\:13-}\\left(\\ln\\left|u\\right|\\right)=\\ln\\left(13\\right)$$", "input": "\\lim_{u\\to\\:13-}\\left(\\ln\\left|u\\right|\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=13$$", "result": "=\\ln\\left|13\\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": "Apply absolute rule: $$\\left|a\\right|=a,\\:a\\ge0$$", "secondary": [ "$$\\left|13\\right|=13$$" ], "result": "=\\ln\\left(13\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\ln\\left(13\\right)-\\ln\\left(5\\right)" } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s79cFAI8gYl5HxBYl8KTTJuXfQ5auaQyfm2QxyWXiiCUAx99rdeE82OzlrOpu9sjdBNZ+USGQPE5+DQyPWBvbsnAPHXkE4uBo0QNWXY+BjOuFndmO9rZiIrtAHYQtY8hBXXql8XXPq6bNQlMm+36iNhmIsAHIkXRe6v0CQ+qFmJtW3T8KO0JtVOqEEoDvY/aXow==" } }, { "type": "step", "result": "=\\frac{1}{2}\\left(\\ln\\left(13\\right)-\\ln\\left(5\\right)\\right)" } ], "meta": { "solvingClass": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }