integral of sqrt(2x-5)

{ "query": { "display": "$$\\int\\:\\sqrt{2x-5}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\sqrt{2x-5}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{1}{3}(2x-5)^{\\frac{3}{2}}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\sqrt{2x-5}dx=\\frac{1}{3}\\left(2x-5\\right)^{\\frac{3}{2}}+C$$", "input": "\\int\\:\\sqrt{2x-5}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\sqrt{2x-5}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: $$u=2x-5$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=2$$", "input": "\\frac{d}{dx}\\left(2x-5\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(2x\\right)-\\frac{d}{dx}\\left(5\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(2x\\right)=2$$", "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+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51jH4j/fzMjnIhJwos1vPNWw" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(5\\right)=0$$", "input": "\\frac{d}{dx}\\left(5\\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+idP5vLVrjUll6eMdYmXEh6/dOKVl5+UiJ6t4qwxJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTvz/OzRy6l5fd6++0L3aMbw" } }, { "type": "step", "result": "=2-0" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{2}du$$" }, { "type": "step", "result": "=\\int\\:\\sqrt{u}\\frac{1}{2}du" }, { "type": "interim", "title": "Simplify $$\\sqrt{u}\\frac{1}{2}:{\\quad}\\frac{\\sqrt{u}}{2}$$", "input": "\\sqrt{u}\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\sqrt{u}}{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\sqrt{u}=\\sqrt{u}$$", "result": "=\\frac{\\sqrt{u}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{\\sqrt{u}}{2}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s79GQHH4KNF/BAwa1rLHu9g+k3hxk9aCfAWodBRxXgUex8EuHmP+Mlx3UWu/2z/XFLDSlYx+wlzV79nT5r/AVmqJV1rB90a1bhpRcesr0OVYt4CyHIOWmjuLQcdHC5MHtIKN6Hv6MoTMtvtU0IQwXdn84k+SM9uK5ZgVIsXdnKy6Jugs8OYsrd0uK6mBpcEuDFg==" } }, { "type": "step", "result": "=\\int\\:\\frac{\\sqrt{u}}{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\\:\\sqrt{u}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\sqrt{u}du", "result": "=\\frac{1}{2}\\cdot\\:\\frac{2}{3}u^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\int\\:u^{\\frac{1}{2}}du", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{u^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}:{\\quad}\\frac{2}{3}u^{\\frac{3}{2}}$$", "input": "\\frac{u^{\\frac{1}{2}+1}}{\\frac{1}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}+1:{\\quad}\\frac{3}{2}$$", "input": "\\frac{1}{2}+1", "result": "=\\frac{u^{\\frac{1}{2}+1}}{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{1}{2}+\\frac{1\\cdot\\:2}{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+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$1+1\\cdot\\:2=3$$", "input": "1+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=1+2" }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JDYTlBE4VraUkLRwnMmR2t6GQqufR6tr2vPxOUv7H++tj2bbSfUlSjnYqC784D/mXx2gq2/8uoBg1ahOTmc2TNo/74bofy09c99TpWFhG0I=" } }, { "type": "step", "result": "=\\frac{3}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "$$u^{\\frac{1}{2}+1}=u^{\\frac{3}{2}}$$", "input": "u^{\\frac{1}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{1}{2}+1:{\\quad}\\frac{3}{2}$$", "input": "\\frac{1}{2}+1", "result": "=u^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{1}{2}+\\frac{1\\cdot\\:2}{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+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$1+1\\cdot\\:2=3$$", "input": "1+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=1+2" }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JDYTlBE4VraUkLRwnMmR2t6GQqufR6tr2vPxOUv7H++tj2bbSfUlSjnYqC784D/mXx2gq2/8uoBg1ahOTmc2TNo/74bofy09c99TpWFhG0I=" } }, { "type": "step", "result": "=\\frac{3}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VcI2MpaClJgyGWg1EkySKSa+StGnWtengtTZdLN8SGHMwViaLUXkeD+JukROhWdjPsK4nwItyc8eEN4jmVbFfP8//6/nV5O4fb8Xgwi7maqZIv8Qu8qROHr+rX3oSxBm0ag3w12wwwjhyGYClc4LNViVI3uvN1by+AN9NfjoKFU=" } }, { "type": "step", "result": "=\\frac{u^{\\frac{3}{2}}}{\\frac{3}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{u^{\\frac{3}{2}}\\cdot\\:2}{3}" }, { "type": "step", "result": "=\\frac{2}{3}u^{\\frac{3}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{2}{3}u^{\\frac{3}{2}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s70DNu1GBaKyufu/4pamubhcsjvX7KVUO/AeCFSId4S33HipIftvBYl8MvlbM/MS0IniX35dQ/h01lIvxamZtt5MkwcYe5THY5CBJSpE9HqGJxESrjjaZRaDGtyZzqQyBzoEFMST8lDZxn1Yq5HMKVTsN/SZgQjH1OoahVjOHG2Hrj8LNU2fafRgGTDrnDOEnog==" } }, { "type": "step", "primary": "Substitute back $$u=2x-5$$", "result": "=\\frac{1}{2}\\cdot\\:\\frac{2}{3}\\left(2x-5\\right)^{\\frac{3}{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\cdot\\:\\frac{2}{3}\\left(2x-5\\right)^{\\frac{3}{2}}:{\\quad}\\frac{1}{3}\\left(2x-5\\right)^{\\frac{3}{2}}$$", "input": "\\frac{1}{2}\\cdot\\:\\frac{2}{3}\\left(2x-5\\right)^{\\frac{3}{2}}", "result": "=\\frac{1}{3}\\left(2x-5\\right)^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=\\frac{1\\cdot\\:2\\left(2x-5\\right)^{\\frac{3}{2}}}{2\\cdot\\:3}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1\\cdot\\:\\left(2x-5\\right)^{\\frac{3}{2}}}{3}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(2x-5\\right)^{\\frac{3}{2}}=\\left(2x-5\\right)^{\\frac{3}{2}}$$", "result": "=\\frac{\\left(2x-5\\right)^{\\frac{3}{2}}}{3}" }, { "type": "step", "result": "=\\frac{1}{3}\\left(2x-5\\right)^{\\frac{3}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8YM8PijhZswI5GTIAEVyDfY91cHu46bu2TSl/WEHcEG4TIQr8/oKLRAj7zS+V3+h2auO77lnnV79llIPeBEfCMGgNzQiLo150r45PvyG40BX3KjoVrvJiWYXh3yjuKNg4Ok+KKUPvS9SYIBRdKMJESmY3ASC+aZqPN1DBWUUsybFLOoCuoLwpnU8v8HDSwilTETUMFgExT938w5nU5x7CmiPwqikc2N0V7RTLGa/18XXIKJTV08A53D8LtjyjJetFQ==" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{1}{3}\\left(2x-5\\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}(2x-5)^{\\frac{3}{2}}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }