integral from 6 to 8 of (88)/((x-6)^3)

{ "query": { "display": "$$\\int_{6}^{8}\\frac{88}{\\left(x-6\\right)^{3}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int _{6}^{8}\\frac{88}{(x-6)^{3}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Definite Integrals", "default": "\\mathrm{diverges}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int_{6}^{8}\\frac{88}{\\left(x-6\\right)^{3}}dx=$$diverges", "input": "\\int_{6}^{8}\\frac{88}{\\left(x-6\\right)^{3}}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": "=88\\cdot\\:\\int_{6}^{8}\\frac{1}{\\left(x-6\\right)^{3}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int_{6}^{8}\\frac{1}{\\left(x-6\\right)^{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: $$u=x-6$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=1$$", "input": "\\frac{d}{dx}\\left(x-6\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{dx}{dx}-\\frac{d}{dx}\\left(6\\right)" }, { "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": "interim", "title": "$$\\frac{d}{dx}\\left(6\\right)=0$$", "input": "\\frac{d}{dx}\\left(6\\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+idP5vLVrjUll6eMdYg8p7Gq8hcikAAMclWLaxZJJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtyoiJomQLyoKTDK4FJPEzd" } }, { "type": "step", "result": "=1-0" }, { "type": "step", "primary": "Simplify", "result": "=1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=1dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=1du$$" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u^{3}}\\cdot\\:1du" }, { "type": "step", "result": "=\\int\\:\\frac{1}{u^{3}}du" }, { "type": "step", "primary": "Adjust integral boundaries:" }, { "type": "interim", "title": "$$x=6\\quad\\Rightarrow\\:u=0$$", "input": "u=x-6", "steps": [ { "type": "step", "primary": "Plug in $$x=6$$", "result": "=6-6" }, { "type": "step", "primary": "Subtract the numbers: $$6-6=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PRVNULXRVZMJ2O1CwcveqQlAlm5MBjmOz6iqN1PySyUpc+uBSermtGr0K/DVx4Ny2tXko4kX7CExiUlL/Q43xMoIDhBIFw/eqhACKhsp95Ikt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$x=8\\quad\\Rightarrow\\:u=2$$", "input": "u=x-6", "steps": [ { "type": "step", "primary": "Plug in $$x=8$$", "result": "=8-6" }, { "type": "step", "primary": "Subtract the numbers: $$8-6=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PRVNULXRVZMJ2O1CwcveqQlAlm5MBjmOz6iqN1PySyUD/eZyMbc7Y59bWW14gxq1KR8JbPr3lCLF3qVgqp4il2OzVH65g5iLfLKFoOyZxDMkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\int_{0}^{2}\\frac{1}{u^{3}}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7whUhkmvgdzlAhmHoX6xJsOmPnL+YARzOraKWbhXQfcNpN4cZPWgnwFqHQUcV4FHsfBLh5j/jJcd1Frv9s/1xSw0pWMfsJc1e/Z0+a/wFZqi4vHnmMpLsSIHO3k0MuLub88q6NoDgGGF5LW1wAR3qe71GQx0UVZKwzkKU2UzVuqKUgb+os4Xi1VCkP3pqJAFMhCguEpg34HWgdIfz1pVC5uwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=88\\cdot\\:\\int_{0}^{2}\\frac{1}{u^{3}}du" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int_{0}^{2}\\frac{1}{u^{3}}du", "result": "=88[-\\frac{1}{2u^{2}}]_{0}^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\frac{1}{a^b}=a^{-b}$$", "secondary": [ "$$\\frac{1}{u^{3}}=u^{-3}$$" ], "result": "=\\int_{0}^{2}u^{-3}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^{-3+1}}{-3+1}]_{0}^{2}" }, { "type": "interim", "title": "Simplify $$\\frac{u^{-3+1}}{-3+1}:{\\quad}-\\frac{1}{2u^{2}}$$", "input": "\\frac{u^{-3+1}}{-3+1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-3+1=-2$$", "result": "=\\frac{u^{-2}}{-2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{u^{-2}}{2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$", "secondary": [ "$$u^{-2}=\\frac{1}{u^{2}}$$" ], "result": "=-\\frac{\\frac{1}{u^{2}}}{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "secondary": [ "$$\\frac{\\frac{1}{u^{2}}}{2}=\\frac{1}{u^{2}\\cdot\\:2}$$" ], "result": "=-\\frac{1}{u^{2}\\cdot\\:2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=[-\\frac{1}{2u^{2}}]_{0}^{2}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s75VaqesnygVEFhTMz5ynuzH80sO/SVnnUhjZoTtNMS57B42dCZngAt2UhJjMVkTWLrb0wH5K3jh3Nl++vthvFt6QyLm3V++zCVSUr02cIsrX/NGC+3s4c0VUWvCmsl0hWuOTSQdnvoQa6lkulr3WAUfwt9LEn7QCBUukJKctfSJK4lcsLSwWUZyz6kqGPpof52xC1zQtG0dma6Lm39471YM=" } }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}$$diverges", "input": "[-\\frac{1}{2u^{2}}]_{0}^{2}", "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": "$$\\lim_{u\\to\\:0+}\\left(-\\frac{1}{2u^{2}}\\right)=-\\infty\\:$$", "input": "\\lim_{u\\to\\:0+}\\left(-\\frac{1}{2u^{2}}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=-\\frac{1}{2}\\cdot\\:\\lim_{u\\to\\:0+}\\left(\\frac{1}{u^{2}}\\right)" }, { "type": "step", "primary": "For $$u\\:$$approaching $$0\\:$$from the right$$,\\:u>0\\quad\\Rightarrow\\quad\\:u^{2}>0$$", "secondary": [ "The denominator is a positive quantity approaching 0 from the right" ], "result": "=-\\frac{1}{2}\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$-c\\cdot\\infty=-\\infty$$", "result": "=-\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{u\\to\\:2-}\\left(-\\frac{1}{2u^{2}}\\right)=-\\frac{1}{8}$$", "input": "\\lim_{u\\to\\:2-}\\left(-\\frac{1}{2u^{2}}\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=2$$", "result": "=-\\frac{1}{2\\cdot\\:2^{2}}", "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": "interim", "title": "Simplify $$-\\frac{1}{2\\cdot\\:2^{2}}:{\\quad}-\\frac{1}{8}$$", "input": "-\\frac{1}{2\\cdot\\:2^{2}}", "result": "=-\\frac{1}{8}", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:2^{2}=2^{3}$$", "input": "2\\cdot\\:2^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$2\\cdot\\:2^{2}=\\:2^{1+2}$$" ], "result": "=2^{1+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=2^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7q8IcoBBuierC2qIyIn9dBy061ljBSPJeENOw2efoSWu4r1cTHNz0uwddWn9KBu6s/z//r+dXk7h9vxeDCLuZqtaOstNvPv01Ty45YNRlaXDII+mBznc+g6xGnMwA8n4l" } }, { "type": "step", "result": "=-\\frac{1}{2^{3}}" }, { "type": "step", "primary": "$$2^{3}=8$$", "result": "=-\\frac{1}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7umP6IcGM/wxZZ+4zn0BXJdtiB/sLvNm95qkvvPliNYzdd47a0hQ8flDbGsI5To1dsabmmR9ZBQ+yLvQJaWqH0OkaQtAPQrPisQY31lThqwiLGmNnLPWGf9PH3lpmjoJIRFLH/GB8eyzJ/fG9m0arlHLiNYoLhAdu4sdcRtbD5qaJqVxX90jlMfh9fKn6dzC4" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\mathrm{diverges}" }, { "type": "step", "primary": "Since $$\\lim_{u\\to\\:0+}\\left(-\\frac{1}{2u^{2}}\\right)=-\\infty\\:$$", "result": "=\\mathrm{diverges}" } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77lozyvIURBhUNMI10ro7nQUae2XCcbkLasiQ4YFoovPJvVfwZNYm42nsliYYK8ONf8NXDgXie/QV5DGH9iLEoxPOlVCLuLF8/mSIHfINeR9Sik02VSIoB32aDhmcD3m0E3kCh3oevUunZ7/b0qFKBQa8jNgCcLmdZ5YPnz1v2WkutwZu5d6xOTx6/K/RRRbpA==" } }, { "type": "step", "result": "=\\mathrm{diverges}" } ], "meta": { "solvingClass": "Integrals" } }, "meta": { "showVerify": true } }