integral from 2 to 3 of (28)/(sqrt(3-x))

{ "query": { "display": "$$\\int_{2}^{3}\\frac{28}{\\sqrt{3-x}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int _{2}^{3}\\frac{28}{\\sqrt{3-x}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Definite Integrals", "default": "56", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int_{2}^{3}\\frac{28}{\\sqrt{3-x}}dx=56$$", "input": "\\int_{2}^{3}\\frac{28}{\\sqrt{3-x}}dx", "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": "=28\\cdot\\:\\int_{2}^{3}\\frac{1}{\\sqrt{3-x}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int_{2}^{3}\\frac{1}{\\sqrt{3-x}}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: $$u=3-x$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=-1$$", "input": "\\frac{d}{dx}\\left(3-x\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(3\\right)-\\frac{dx}{dx}" }, { "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": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "step", "result": "=0-1" }, { "type": "step", "primary": "Simplify", "result": "=-1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=-1dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\left(-1\\right)du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{\\sqrt{u}}\\left(-1\\right)du" }, { "type": "step", "result": "=\\int\\:-\\frac{1}{\\sqrt{u}}du" }, { "type": "step", "primary": "Adjust integral boundaries:" }, { "type": "interim", "title": "$$x=2\\quad\\Rightarrow\\:u=1$$", "input": "u=3-x", "steps": [ { "type": "step", "primary": "Plug in $$x=2$$", "result": "=3-2" }, { "type": "step", "primary": "Subtract the numbers: $$3-2=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72fxortQ/4xLQ+NoCt3xzJQlAlm5MBjmOz6iqN1PySyXNsbio4EtCJeiTO1yjhzmju1lwM9F5DGqpHl6pj7ybq2kAb6lH5ObwfzaD8YREw9Ekt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$x=3\\quad\\Rightarrow\\:u=0$$", "input": "u=3-x", "steps": [ { "type": "step", "primary": "Plug in $$x=3$$", "result": "=3-3" }, { "type": "step", "primary": "Subtract the numbers: $$3-3=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72fxortQ/4xLQ+NoCt3xzJQlAlm5MBjmOz6iqN1PySyUpc+uBSermtGr0K/DVx4NyNCK5IlamY8658jIJQWq5YcoIDhBIFw/eqhACKhsp95Ikt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\int_{1}^{0}-\\frac{1}{\\sqrt{u}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s79f5nbg0/LpQl8zO0P3Y8SKkuMV6lnzdABBve2OBkhlnHI5S0StY1FdtOqqOPr0Te8kZykkSQR7JxF6tcEv88MO4fClF/YTdtcCRNeh/zX2c2K6z/VeWlqrHefxr8N6AL2ZYxFotSmZrw4o69TMioBp2seJ/EMmTR7hO4WtIsKVK1sD7NfhsPe7eDHrmjY0mE4/yM86NQA4MIhKMbcQ9RSwohx3JNTaptNuNW2szzOqF" } }, { "type": "step", "result": "=28\\cdot\\:\\int_{1}^{0}-\\frac{1}{\\sqrt{u}}du" }, { "type": "step", "primary": "$$\\int_{a}^{b}f\\left(x\\right)dx=-\\int_{b}^{a}f\\left(x\\right)dx,\\:a<b$$", "result": "=28\\left(-\\int_{0}^{1}-\\frac{1}{\\sqrt{u}}du\\right)" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=28\\left(-\\left(-\\int_{0}^{1}\\frac{1}{\\sqrt{u}}du\\right)\\right)" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{0}^{1}\\frac{1}{\\sqrt{u}}du", "result": "=28\\left(-\\left(-[2u^{\\frac{1}{2}}]_{0}^{1}\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\int_{0}^{1}\\frac{1}{u^{\\frac{1}{2}}}du", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{u^{\\frac{1}{2}}}=u^{-\\frac{1}{2}}$$" ], "result": "=\\int_{0}^{1}u^{-\\frac{1}{2}}du", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "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}]_{0}^{1}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{-\\frac{1}{2}+1}}{-\\frac{1}{2}+1}:{\\quad}2u^{\\frac{1}{2}}$$", "input": "\\frac{u^{-\\frac{1}{2}+1}}{-\\frac{1}{2}+1}", "steps": [ { "type": "interim", "title": "Join $$-\\frac{1}{2}+1:{\\quad}\\frac{1}{2}$$", "input": "-\\frac{1}{2}+1", "result": "=\\frac{u^{-\\frac{1}{2}+1}}{\\frac{1}{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": "$$-1+1\\cdot\\:2=1$$", "input": "-1+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-1+2" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+2=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vn6CQNwxYfV8gDG5KuxmLlXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qk4XEODzDfV3RCKyB8F0QybUZ0qCI2z4+iUq9O5bNiT8=" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } }, { "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}{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": "$$-1+1\\cdot\\:2=1$$", "input": "-1+1\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=-1+2" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+2=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vn6CQNwxYfV8gDG5KuxmLlXTSum/z5kLpMzXS1UJIexiYFOaxxQqg3o8CNKkTm2qk4XEODzDfV3RCKyB8F0QybUZ0qCI2z4+iUq9O5bNiT8=" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Join Concise Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gmcjsTVlHGpoV2sBpqUTxqL3R/dXk768t6JSjGXJZIJwkKGJWEPFPk38sdJMsyPIlO4GlLPYnU249VRH/lBBCgH2kDe5DGYTz3TrPquGdIjIpABNda1kOooFpT/11VT6NqwMC9Xx4WE2eLLY/kBb/1QW3Chm7McvYpuS87Y5EFs=" } }, { "type": "step", "result": "=\\frac{u^{\\frac{1}{2}}}{\\frac{1}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{u^{\\frac{1}{2}}\\cdot\\:2}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=2u^{\\frac{1}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=[2u^{\\frac{1}{2}}]_{0}^{1}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s78UmysLuBDtZMioOquwC8SNKA7gCa+xzDISCnOJ25VCRpN4cZPWgnwFqHQUcV4FHsS92UCn2Gr7vCBg6Uxy6Qx3jgT/SrQIc7mFzkmCezRK3xdcL161ubEiuNt7qcWkvnGqUNdTsBnk30r8s6qKZZCRFKk3fejFkyiOiq9iG9IkAW4E2R7R5kA2ZAx0+pWxTUzmaCTF+08KtdZkgx0pZ/mA=" } }, { "type": "interim", "title": "Simplify $$28\\left(-\\left(-[2u^{\\frac{1}{2}}]_{0}^{1}\\right)\\right):{\\quad}28[2\\sqrt{u}]_{0}^{1}$$", "input": "28\\left(-\\left(-[2u^{\\frac{1}{2}}]_{0}^{1}\\right)\\right)", "result": "=28[2\\sqrt{u}]_{0}^{1}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=28[2u^{\\frac{1}{2}}]_{0}^{1}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "result": "=28[2\\sqrt{u}]_{0}^{1}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tWdrydfa7dRFoHf3JPwUQnQD5tX6MastIpQqEKc+kae3jI4Odj0JRLgVreRmQWB/cJChiVhDxT5N/LHSTLMjyH5xQ4jHWv+t62et5aVL0rpl0zdt6VnFpdi+9HFIT1ZmeqXxdc+rps1CUyb7fqI2GYMB7YweilrASoE1JqE8kZq0i2cXxrDhQeYg/8kbrCVwCqMwPEw4swbCw3yuiXCf1yS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}2$$", "input": "[2\\sqrt{u}]_{0}^{1}", "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": "$$\\lim_{u\\to\\:0+}\\left(2\\sqrt{u}\\right)=0$$", "input": "\\lim_{u\\to\\:0+}\\left(2\\sqrt{u}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=0$$", "result": "=2\\sqrt{0}", "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": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{u\\to\\:1-}\\left(2\\sqrt{u}\\right)=2$$", "input": "\\lim_{u\\to\\:1-}\\left(2\\sqrt{u}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=1$$", "result": "=2\\sqrt{1}", "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": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=2-0" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74qU6WE+xz88IQUkaKhvS+tRaqoMO2RHVzA2RopSx+pa/koYGDkQ4SWDRIfahfSErLgcEK0UaIjA3sJ8EGJLskc6acXjldgP2e4Nh6wgdBWt/z//r+dXk7h9vxeDCLuZqlP7XdP66vn2lDIgOgst4mRdapHA9w5BUr5XcUy5iXqQialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "result": "=28\\cdot\\:2" }, { "type": "step", "primary": "Simplify", "result": "=56" } ], "meta": { "solvingClass": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }