integral of-tsqrt(t-1)

{ "query": { "display": "$$\\int\\:-t\\sqrt{t-1}dt$$", "symbolab_question": "BIG_OPERATOR#\\int -t\\sqrt{t-1}dt" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\frac{2}{5}(t-1)^{\\frac{5}{2}}-\\frac{2}{3}(t-1)^{\\frac{3}{2}}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:-t\\sqrt{t-1}dt=-\\frac{2}{5}\\left(t-1\\right)^{\\frac{5}{2}}-\\frac{2}{3}\\left(t-1\\right)^{\\frac{3}{2}}+C$$", "input": "\\int\\:-t\\sqrt{t-1}dt", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=-\\int\\:t\\sqrt{t-1}dt" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:t\\sqrt{t-1}dt", "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: $$u=t-1$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dt}=1$$", "input": "\\frac{d}{dt}\\left(t-1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dt}{dt}-\\frac{d}{dt}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{dt}{dt}=1$$", "input": "\\frac{dt}{dt}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dt}{dt}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuWV6zCUVy7FvtVpq63L1y1jqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIUVox5V37RRgiM2tHP1hZLC9fXkG27pZ636yeVofyg8V" } }, { "type": "interim", "title": "$$\\frac{d}{dt}\\left(1\\right)=0$$", "input": "\\frac{d}{dt}\\left(1\\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+idP5vLVrjUll6eMdYsXHPBaM8/e7W8DZUAY0LNZJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTuDmriYSpovrBcd7ideomaa" } }, { "type": "step", "result": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=1dt$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dt=1du$$" }, { "type": "step", "result": "=\\int\\:t\\sqrt{u}\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:t\\sqrt{u}du" }, { "type": "interim", "title": "$$u=t-1\\quad\\Rightarrow\\quad\\:t=u+1$$", "input": "t-1=u", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "t-1=u", "result": "t=u+1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "t-1+1=u+1" }, { "type": "step", "primary": "Simplify", "result": "t=u+1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\left(u+1\\right)\\sqrt{u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7w3vYgzeF0DX0i5PS9VgzTek3hxk9aCfAWodBRxXgUex8EuHmP+Mlx3UWu/2z/XFLDSlYx+wlzV79nT5r/AVmqLUujGroJmE8t1YGpgBiR2TsUKWqbSJvVDvHbKmg5DURnql8XXPq6bNQlMm+36iNhkkjuzIgeJUg10ybKgq0r22txEId7lZcSHdTAsAvmTZFg==" } }, { "type": "step", "result": "=-\\int\\:\\left(u+1\\right)\\sqrt{u}du" }, { "type": "interim", "title": "Expand $$\\left(u+1\\right)\\sqrt{u}:{\\quad}u^{\\frac{3}{2}}+\\sqrt{u}$$", "input": "\\left(u+1\\right)\\sqrt{u}", "steps": [ { "type": "step", "result": "=\\sqrt{u}\\left(u+1\\right)" }, { "type": "step", "primary": "Apply the distributive law: $$a\\left(b+c\\right)=ab+ac$$", "secondary": [ "$$a=\\sqrt{u},\\:b=u,\\:c=1$$" ], "result": "=\\sqrt{u}u+\\sqrt{u}\\cdot\\:1", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "result": "=u\\sqrt{u}+1\\cdot\\:\\sqrt{u}" }, { "type": "interim", "title": "Simplify $$u\\sqrt{u}+1\\cdot\\:\\sqrt{u}:{\\quad}u^{\\frac{3}{2}}+\\sqrt{u}$$", "input": "u\\sqrt{u}+1\\cdot\\:\\sqrt{u}", "result": "=u^{\\frac{3}{2}}+\\sqrt{u}", "steps": [ { "type": "interim", "title": "$$u\\sqrt{u}=u^{\\frac{3}{2}}$$", "input": "u\\sqrt{u}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$u\\sqrt{u}=\\:uu^{\\frac{1}{2}}=\\:u^{1+\\frac{1}{2}}$$" ], "result": "=u^{1+\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "Join $$1+\\frac{1}{2}:{\\quad}\\frac{3}{2}$$", "input": "1+\\frac{1}{2}", "result": "=u^{\\frac{3}{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": "$$1\\cdot\\:2+1=3$$", "input": "1\\cdot\\:2+1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=2+1" }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73KRcHAW/izwlTVvOREpZp96GQqufR6tr2vPxOUv7H++tj2bbSfUlSjnYqC784D/mP/n/sT8Hudl/0KJRqY9qeT5m7NvIHd7cmyKBlstyHmM=" } }, { "type": "step", "result": "=\\frac{3}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7jFv7Hkp4pRtCa9pk4sCzGCAn9lkDfZkicUGkO3EF+Irbzc71ZPyJR8svQzMB156o22e9RboyPv7NrBwjunrnS8EFNa4juybgdm8o+gCr+zOnnJwZq0sxzCtordGTjrBnWJUje683VvL4A301+OgoVQ==" } }, { "type": "interim", "title": "$$1\\cdot\\:\\sqrt{u}=\\sqrt{u}$$", "input": "1\\cdot\\:\\sqrt{u}", "steps": [ { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\sqrt{u}=\\sqrt{u}$$", "result": "=\\sqrt{u}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UHFhuJ6yJNDOcGH7Fz25/1OtvOxF9XLHJjk1uV6GRpijkVi15I8rBefLi4Iyt2wrFkj4MhgjnafuNQk07WVfSjya2IcHPs77h6zi8+fDJdogAtott3NXJo018yTJN+UisxO/W69xaAr3ciqtIs7XKA==" } }, { "type": "step", "result": "=u^{\\frac{3}{2}}+\\sqrt{u}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7f8si265VkEppndhTZ6R+8ACWKUbvV6WK3fDUgFtg3Q+1p3KlYTIciqkhJ79+uU3uMF2FYlm3v4VyuY6XRyXSKdn6oxbMODVt/JZrhS+cU8JN5Aod6Hr1Lp2e/29KhSgUglsvCEouUOqkYmHP7w+QkTUNzz++FaHiPDxM0pOYTyo=" } }, { "type": "step", "result": "=-\\int\\:u^{\\frac{3}{2}}+\\sqrt{u}du" }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=-\\left(\\int\\:u^{\\frac{3}{2}}du+\\int\\:\\sqrt{u}du\\right)" }, { "type": "interim", "title": "$$\\int\\:u^{\\frac{3}{2}}du=\\frac{2}{5}u^{\\frac{5}{2}}$$", "input": "\\int\\:u^{\\frac{3}{2}}du", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:u^{\\frac{3}{2}}du", "result": "=\\frac{2}{5}u^{\\frac{5}{2}}", "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^{\\frac{3}{2}+1}}{\\frac{3}{2}+1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{\\frac{3}{2}+1}}{\\frac{3}{2}+1}:{\\quad}\\frac{2}{5}u^{\\frac{5}{2}}$$", "input": "\\frac{u^{\\frac{3}{2}+1}}{\\frac{3}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{3}{2}+1:{\\quad}\\frac{5}{2}$$", "input": "\\frac{3}{2}+1", "result": "=\\frac{u^{\\frac{3}{2}+1}}{\\frac{5}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{3}{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{3+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$3+1\\cdot\\:2=5$$", "input": "3+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=3+2" }, { "type": "step", "primary": "Add the numbers: $$3+2=5$$", "result": "=5" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77WYQVM/BFjf2+3WojNhh9N6GQqufR6tr2vPxOUv7H+9Ao0ShslLSdH/VCTigfQH+bpXOvZCwJCiCuwTrm2fYMRM4RxLmKFZNkO9PZpS9A0U=" } }, { "type": "step", "result": "=\\frac{5}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "type": "interim", "title": "$$u^{\\frac{3}{2}+1}=u^{\\frac{5}{2}}$$", "input": "u^{\\frac{3}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$\\frac{3}{2}+1:{\\quad}\\frac{5}{2}$$", "input": "\\frac{3}{2}+1", "result": "=u^{\\frac{5}{2}}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:2}{2}$$", "result": "=\\frac{3}{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{3+1\\cdot\\:2}{2}" }, { "type": "interim", "title": "$$3+1\\cdot\\:2=5$$", "input": "3+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=3+2" }, { "type": "step", "primary": "Add the numbers: $$3+2=5$$", "result": "=5" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77WYQVM/BFjf2+3WojNhh9N6GQqufR6tr2vPxOUv7H+9Ao0ShslLSdH/VCTigfQH+bpXOvZCwJCiCuwTrm2fYMRM4RxLmKFZNkO9PZpS9A0U=" } }, { "type": "step", "result": "=\\frac{5}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fveYAFpxUtSQwLHqQlXg9Ca+StGnWtengtTZdLN8SGHMwViaLUXkeD+JukROhWdjRLiiskflxXRfwmCuIHY0//8//6/nV5O4fb8Xgwi7maqMZWGaN/++dCOmxIcAgmRik6JxDgkBVsvNoLBDj9XAMViVI3uvN1by+AN9NfjoKFU=" } }, { "type": "step", "result": "=\\frac{u^{\\frac{5}{2}}}{\\frac{5}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{u^{\\frac{5}{2}}\\cdot\\:2}{5}" }, { "type": "step", "result": "=\\frac{2}{5}u^{\\frac{5}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{2}{5}u^{\\frac{5}{2}}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7zSdEdZ9bat0ApgKqviwHE0y4+rY5ULRUEksemusM4Yyrrf9ZAnPXwtHEGeHjeiUc8XwLUgD2yVoFe9iCfntTx6JB7Asf8mi7IuDtv3AUYazlCqDsnnaioCqjNKu3ETMHfmUz1GYa4snANpQYMxmVABSBv6izheLVUKQ/emokAUyVi4djWqB/aQf/oQfiXX55rCI2sSeA74029n2yo277ZU=" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\sqrt{u}du=\\frac{2}{3}u^{\\frac{3}{2}}$$", "input": "\\int\\:\\sqrt{u}du", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:\\sqrt{u}du", "result": "=\\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==" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-\\left(\\frac{2}{5}u^{\\frac{5}{2}}+\\frac{2}{3}u^{\\frac{3}{2}}\\right)" }, { "type": "step", "primary": "Substitute back $$u=t-1$$", "result": "=-\\left(\\frac{2}{5}\\left(t-1\\right)^{\\frac{5}{2}}+\\frac{2}{3}\\left(t-1\\right)^{\\frac{3}{2}}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=-\\frac{2}{5}\\left(t-1\\right)^{\\frac{5}{2}}-\\frac{2}{3}\\left(t-1\\right)^{\\frac{3}{2}}", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\frac{2}{5}\\left(t-1\\right)^{\\frac{5}{2}}-\\frac{2}{3}\\left(t-1\\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": "t", "plotRequest": "y=-\\frac{2}{5}(t-1)^{\\frac{5}{2}}-\\frac{2}{3}(t-1)^{\\frac{3}{2}}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }