integral of (xsqrt(x^2+3))

{ "query": { "display": "$$\\int\\:\\left(x\\sqrt{x^{2}+3}\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int (x\\sqrt{x^{2}+3})dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{1}{3}(x^{2}+3)^{\\frac{3}{2}}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\left(x\\sqrt{x^{2}+3}\\right)dx=\\frac{1}{3}\\left(x^{2}+3\\right)^{\\frac{3}{2}}+C$$", "input": "\\int\\:x\\sqrt{x^{2}+3}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:x\\sqrt{x^{2}+3}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: $$u=\\sqrt{x^{2}+3}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=\\frac{x}{\\sqrt{x^{2}+3}}$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x^{2}+3}\\right)", "steps": [ { "type": "interim", "title": "Apply the chain rule:$${\\quad}\\frac{1}{2\\sqrt{x^{2}+3}}\\frac{d}{dx}\\left(x^{2}+3\\right)$$", "input": "\\frac{d}{dx}\\left(\\sqrt{x^{2}+3}\\right)", "result": "=\\frac{1}{2\\sqrt{x^{2}+3}}\\frac{d}{dx}\\left(x^{2}+3\\right)", "steps": [ { "type": "step", "primary": "Apply the chain rule: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$", "secondary": [ "$$f=\\sqrt{u},\\:\\:u=x^{2}+3$$" ], "result": "=\\frac{d}{du}\\left(\\sqrt{u}\\right)\\frac{d}{dx}\\left(x^{2}+3\\right)", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule", "practiceTopic": "Chain Rule" } }, { "type": "interim", "title": "$$\\frac{d}{du}\\left(\\sqrt{u}\\right)=\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{d}{du}\\left(\\sqrt{u}\\right)", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\frac{d}{du}\\left(u^{\\frac{1}{2}}\\right)", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=\\frac{1}{2}u^{\\frac{1}{2}-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}u^{\\frac{1}{2}-1}:{\\quad}\\frac{1}{2\\sqrt{u}}$$", "input": "\\frac{1}{2}u^{\\frac{1}{2}-1}", "result": "=\\frac{1}{2\\sqrt{u}}", "steps": [ { "type": "interim", "title": "$$u^{\\frac{1}{2}-1}=u^{-\\frac{1}{2}}$$", "input": "u^{\\frac{1}{2}-1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}-1:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{1}{2}-1", "result": "=u^{-\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=-\\frac{1\\cdot\\:2}{2}+\\frac{1}{2}" }, { "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\\cdot\\:2+1}{2}" }, { "type": "interim", "title": "$$-1\\cdot\\:2+1=-1$$", "input": "-1\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-2+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-2+1=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s731snK5z/nd3Sq/6JpCqiX1XTSum/z5kLpMzXS1UJIew02FKSBoQo9V3G05AlnWtTyCE30rzMlUAIVDyhseMBropKGn5MuXZnb2ZCo/hVsBU=" } }, { "type": "step", "result": "=\\frac{-1}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VcI2MpaClJgyGWg1EkySKe0se7vRyav6BwUCJZptwG3MwViaLUXkeD+JukROhWdjMvOxDqXzE3/CFO0TFmffHAH2kDe5DGYTz3TrPquGdIhyukSOA/1RgMKO0TMhInPOMabdUggEogUL9RT7PNKh0VQW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{1}{2}u^{-\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{u}}$$" ], "result": "=\\frac{1}{2}\\cdot\\:\\frac{1}{\\sqrt{u}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{1\\cdot\\:1}{2\\sqrt{u}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:1=1$$", "result": "=\\frac{1}{2\\sqrt{u}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JOPQ2g2GS9EQptV8nckZSrH6E/qPf7AlxQDX8MXU5OsAlilG71elit3w1IBbYN0P8rEus7TgCihQBF5omOFkJv2RkT96g5Q5jVbn5fyeQzwB9pA3uQxmE8906z6rhnSIHimBRYRqHSWeJkuUPhfTC1O468YRFxaQeTFqgRqR2rvsVWktCxa7XSYzIK90x3+aTk5AXTHU+C+TrGKWzqT97A==" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{u}}\\frac{d}{dx}\\left(x^{2}+3\\right)" }, { "type": "step", "primary": "Substitute back $$u=x^{2}+3$$", "result": "=\\frac{1}{2\\sqrt{x^{2}+3}}\\frac{d}{dx}\\left(x^{2}+3\\right)" } ], "meta": { "interimType": "Derivative Chain Rule 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjasHTJ4Bo7yMP4qgzIlGERgh6Fj+VHSvChJ0bLgkQuZZ3GoG6Ko8jDPh4vymhs0+tlv8YVMwh/df5SMAfAmpJXv++bSprT8DRLjDQza+XRVtP4dW/8YT8X7GJ2F6L9zL33ga41RLI01/GV+O/PuAiBq6jyoO0530U8ik9DXRQJjRSpN33oxZMojoqvYhvSJAFuBNke0eZANmQMdPqVsU1Pq9PUUBMmxzxm22kIDr81B" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}+3\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}+3\\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(3\\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": "$$\\frac{d}{dx}\\left(3\\right)=0$$", "input": "\\frac{d}{dx}\\left(3\\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+idP5vLVrjUll6eMdYu6nPER/cBcxgb/Kz63vQV1J8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTuwXg0Wd+I5tymlezl5JoPF" } }, { "type": "step", "result": "=2x+0" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "=\\frac{1}{2\\sqrt{x^{2}+3}}\\cdot\\:2x" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2\\sqrt{x^{2}+3}}\\cdot\\:2x:{\\quad}\\frac{x}{\\sqrt{x^{2}+3}}$$", "input": "\\frac{1}{2\\sqrt{x^{2}+3}}\\cdot\\:2x", "result": "=\\frac{x}{\\sqrt{x^{2}+3}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2x}{2\\sqrt{x^{2}+3}}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1\\cdot\\:x}{\\sqrt{x^{2}+3}}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:x=x$$", "result": "=\\frac{x}{\\sqrt{x^{2}+3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WW9Djej2t2V9XX+YqO13C7lOpShcVJEqLrkgjTmQdbptOYrMhCJMBBPrwsYfxOuCq47vuWedXv2WUg94ER8IwYQO7+z45GqFJXp4sUC1pO37asPjgkn3h9UFO6uMOQamRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6oHIcXcImz+eH9oVkJvPejdwa+PYEA3M3at4GyvhZ5ki6zTLYCkr3cyMfwv5DlY8ks=" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{x}{\\sqrt{x^{2}+3}}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{\\sqrt{x^{2}+3}}{x}du$$" }, { "type": "step", "result": "=\\int\\:xu\\frac{\\sqrt{x^{2}+3}}{x}du" }, { "type": "step", "primary": "$$u=\\sqrt{x^{2}+3}$$", "result": "=\\int\\:xu\\frac{u}{x}du" }, { "type": "interim", "title": "Simplify $$xu\\frac{u}{x}:{\\quad}u^{2}$$", "input": "xu\\frac{u}{x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{uxu}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=uu" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$uu=\\:u^{1+1}$$" ], "result": "=u^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:u^{2}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s73vCedvXETBFLXTcJn8i5Q6r+0zX0wqSmAVtc7NV8L0A/6umhM84nJGixAkjaEU36WyJTZPfTsR1UcTUz/dkJr3nlajb5gZ0R8BiMrkcGLKL5tHAXmAzw8wpd3QFM2abuAS4M5VpC8qh+oehjmM1qmxw3obfALRo06lfcYy06xaEialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "result": "=\\int\\:u^{2}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:u^{2}du", "result": "=\\frac{u^{3}}{3}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{u^{2+1}}{2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{2+1}}{2+1}:{\\quad}\\frac{u^{3}}{3}$$", "input": "\\frac{u^{2+1}}{2+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=\\frac{u^{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{u^{3}}{3}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7ztMnkJqJUs4Qp2osi+oqZ+o/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7ofEE1c1KRzl1GtEOEJ8QtOggQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "step", "primary": "Substitute back $$u=\\sqrt{x^{2}+3}$$", "result": "=\\frac{\\left(\\sqrt{x^{2}+3}\\right)^{3}}{3}" }, { "type": "interim", "title": "Simplify $$\\frac{\\left(\\sqrt{x^{2}+3}\\right)^{3}}{3}:{\\quad}\\frac{1}{3}\\left(x^{2}+3\\right)^{\\frac{3}{2}}$$", "input": "\\frac{\\left(\\sqrt{x^{2}+3}\\right)^{3}}{3}", "result": "=\\frac{1}{3}\\left(x^{2}+3\\right)^{\\frac{3}{2}}", "steps": [ { "type": "interim", "title": "$$\\left(\\sqrt{x^{2}+3}\\right)^{3}=\\left(x^{2}+3\\right)^{\\frac{3}{2}}$$", "input": "\\left(\\sqrt{x^{2}+3}\\right)^{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(\\left(x^{2}+3\\right)^{\\frac{1}{2}}\\right)^{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=\\left(x^{2}+3\\right)^{\\frac{1}{2}\\cdot\\:3}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:3=\\frac{3}{2}$$", "input": "\\frac{1}{2}\\cdot\\:3", "result": "=\\left(x^{2}+3\\right)^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{2}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8SPpFLppSOOOwknMsNOrv36rju+5Z51e/ZZSD3gRHwjBnpkZpWG0liIE8buj+sqXg2RLd2VwIqlBNByF6663sySF76Eydb/wnfqEDL8zJUMWNJdlTvBkWjQ+TKBgKD6qgLCI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WpspHYEREl+7q/RNTrR14ZG5/DzV/OhZbn1NwXr1rN0JQJZuTAY5js+oqjdT8kslDQx0i02IwCJXS00m91r2NMitKFL3HCliOi9IfmyyDMVZCseayr8qL6QXe8h+/kZVpUAK5NDqOEDVKJUIc+dTZ0yGRxcT/qWlDwn/ww1++ctHDLEQpFeecO1BMSPhRMdG" } }, { "type": "step", "result": "=\\frac{\\left(x^{2}+3\\right)^{\\frac{3}{2}}}{3}" }, { "type": "step", "result": "=\\frac{1}{3}\\left(x^{2}+3\\right)^{\\frac{3}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kLn7iC+AOhHStEx/knbw2R+7/IfxshVhdt3lXVEjVtjk9CU8MCWpvAxSxvbI1SlD/aL2Coo0GISQwm8bv5wDifboNMSeFk0FEzgfGXCd+QadOZ2p4mNhqLUfdhIZnLUDas41htHRTP01p4839D55UO5AIz++qluupTlLFEcE9J35ApwMgUXeAfnPTPBAhXrHyd0a6XaVbGcYXaMvCZ4njMwWmibBzuwvkb5vMafEDWE=" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{1}{3}\\left(x^{2}+3\\right)^{\\frac{3}{2}}+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=Substitution", "practiceTopic": "Integral Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=\\frac{1}{3}(x^{2}+3)^{\\frac{3}{2}}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }