integral from 0 to 1 of x^2cos(x^3)

{ "query": { "display": "$$\\int_{0}^{1}x^{2}\\cos\\left(x^{3}\\right)dx$$", "symbolab_question": "BIG_OPERATOR#\\int _{0}^{1}x^{2}\\cos(x^{3})dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Definite Integrals", "default": "\\frac{1}{3}\\sin(1)", "decimal": "0.28049…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int_{0}^{1}x^{2}\\cos\\left(x^{3}\\right)dx=\\frac{1}{3}\\sin\\left(1\\right)$$", "input": "\\int_{0}^{1}x^{2}\\cos\\left(x^{3}\\right)dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int_{0}^{1}\\cos\\left(x^{3}\\right)x^{2}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: $$u=x^{3}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=3x^{2}$$", "input": "\\frac{d}{dx}\\left(x^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtb6j95rHG7YtZ73Xx2qCjqk3hxk9aCfAWodBRxXgUexf7nh0v5ML3fMP9GgRVbRX/8//6/nV5O4fb8Xgwi7mapE1PQFvzh2DTeXDl7hvM6tspMmxu9uOrZheNUM2oeSJw==" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=3x^{2}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{3x^{2}}du$$" }, { "type": "step", "result": "=\\int\\:\\cos\\left(u\\right)x^{2}\\frac{1}{3x^{2}}du" }, { "type": "interim", "title": "Simplify $$\\cos\\left(u\\right)x^{2}\\frac{1}{3x^{2}}:{\\quad}\\frac{\\cos\\left(u\\right)}{3}$$", "input": "\\cos\\left(u\\right)x^{2}\\frac{1}{3x^{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:\\cos\\left(u\\right)x^{2}}{3x^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$x^{2}$$", "result": "=\\frac{1\\cdot\\:\\cos\\left(u\\right)}{3}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\cos\\left(u\\right)=\\cos\\left(u\\right)$$", "result": "=\\frac{\\cos\\left(u\\right)}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{\\cos\\left(u\\right)}{3}du" }, { "type": "step", "primary": "Adjust integral boundaries:" }, { "type": "interim", "title": "$$x=0\\quad\\Rightarrow\\:u=0$$", "input": "u=x^{3}", "steps": [ { "type": "step", "primary": "Plug in $$x=0$$", "result": "=0^{3}" }, { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7//tcZH9A89/3+DySD6X62QOfOVs9mPIqDLV5QIWwt3m4DS9snDRdGFIEJoiNCqQWOjJLhm4arQXwOq+lDYfUMO8WzRdAxDlhD+1MdDyljXKwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$x=1\\quad\\Rightarrow\\:u=1$$", "input": "u=x^{3}", "steps": [ { "type": "step", "primary": "Plug in $$x=1$$", "result": "=1^{3}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7//tcZH9A89/3+DySD6X62QOfOVs9mPIqDLV5QIWwt3mXUrWvRppVP8TTNlqQgP2MDhl3gxaeLOJnqws+C/44uxkN2cZhQ0rJwpjs2bCwZtSwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\int_{0}^{1}\\frac{\\cos\\left(u\\right)}{3}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s79qEJWUWlI+D0gEWSfRSUxb/sJ8wmIjCZ7cApzXzPxL6LI71+ylVDvwHghUiHeEt93Gam9ro5jdhRGQuTP5rGCB4l9+XUP4dNZSL8WpmbbeTHgon5ADS7+UTI5Ieyyd2XMpMzjSg/HRoCKZOmPdsVX71GQx0UVZKwzkKU2UzVuqKUgb+os4Xi1VCkP3pqJAFMhCguEpg34HWgdIfz1pVC5uwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\int_{0}^{1}\\frac{\\cos\\left(u\\right)}{3}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}{3}\\cdot\\:\\int_{0}^{1}\\cos\\left(u\\right)du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\cos\\left(u\\right)du=\\sin\\left(u\\right)$$", "result": "=\\frac{1}{3}[\\sin\\left(u\\right)]_{0}^{1}" }, { "type": "interim", "title": "Compute the boundaries:$${\\quad}\\sin\\left(1\\right)$$", "input": "[\\sin\\left(u\\right)]_{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": "$$\\lim_{u\\to\\:0+}\\left(\\sin\\left(u\\right)\\right)=0$$", "input": "\\lim_{u\\to\\:0+}\\left(\\sin\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=0$$", "result": "=\\sin\\left(0\\right)", "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": "Use the following trivial identity:$${\\quad}\\sin\\left(0\\right)=0$$", "secondary": [ "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" ], "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{u\\to\\:1-}\\left(\\sin\\left(u\\right)\\right)=\\sin\\left(1\\right)$$", "input": "\\lim_{u\\to\\:1-}\\left(\\sin\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Plug in the value $$u=1$$", "result": "=\\sin\\left(1\\right)", "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 $$\\lim_{x\\to{a}}g\\left(x\\right)\\neq0$$" } } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=\\sin\\left(1\\right)-0" }, { "type": "step", "primary": "Simplify", "result": "=\\sin\\left(1\\right)", "meta": { "solvingClass": "Solver" } } ], "meta": { "interimType": "Integral Definite Limit Boundaries 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s72HakCeeL9RMmSJLx4O64+IYvspna43jhrM9uyridhP/j1aPKxxYypVIRLf3NPD6Vtkm3azHEZpcKIi0vkgG909W/7Ta01JiwwK31btOnWUO1sD7NfhsPe7eDHrmjY0mE8IUulKmv/BlT86nZPYwocF5q2klkq57GUo3IMZOy6m4" } }, { "type": "step", "result": "=\\frac{1}{3}\\sin\\left(1\\right)" } ], "meta": { "solvingClass": "Integrals" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "yes" }, "showViewLarger": true } }, "meta": { "showVerify": true } }