integral of (sin^5(ln(x)))/x

{ "query": { "display": "$$\\int\\:\\frac{\\sin^{5}\\left(\\ln\\left(x\\right)\\right)}{x}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{\\sin^{5}(\\ln(x))}{x}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "-\\cos(\\ln(x))+\\frac{2\\cos^{3}(\\ln(x))}{3}-\\frac{\\cos^{5}(\\ln(x))}{5}+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{\\sin^{5}\\left(\\ln\\left(x\\right)\\right)}{x}dx=-\\cos\\left(\\ln\\left(x\\right)\\right)+\\frac{2\\cos^{3}\\left(\\ln\\left(x\\right)\\right)}{3}-\\frac{\\cos^{5}\\left(\\ln\\left(x\\right)\\right)}{5}+C$$", "input": "\\int\\:\\frac{\\sin^{5}\\left(\\ln\\left(x\\right)\\right)}{x}dx", "steps": [ { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\sin^{5}\\left(\\ln\\left(x\\right)\\right)}{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: $$u=\\ln\\left(x\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=\\frac{1}{x}$$", "input": "\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dx}\\left(\\ln\\left(x\\right)\\right)=\\frac{1}{x}$$", "result": "=\\frac{1}{x}" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYhHxrkiFdmQgNsZN21633mEcjlLRK1jUV206qo4+vRN78rEus7TgCihQBF5omOFkJlc0OBMs8qTL4oWnxx62vyRTW26qciuyUBGXQExCUedYC94xYOkpqpIRbTXK8bwDkBTzacgE5U7pBE1AeNi+CKY=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{1}{x}dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=xdu$$" }, { "type": "step", "result": "=\\int\\:\\frac{\\sin^{5}\\left(u\\right)}{x}xdu" }, { "type": "interim", "title": "Simplify $$\\frac{\\sin^{5}\\left(u\\right)}{x}x:{\\quad}\\sin^{5}\\left(u\\right)$$", "input": "\\frac{\\sin^{5}\\left(u\\right)}{x}x", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sin^{5}\\left(u\\right)x}{x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\sin^{5}\\left(u\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\sin^{5}\\left(u\\right)du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s72AfVwsUp8BpAFzYp818Z637kB/OghBOO9lqMJo5EirTk3WldPTzMRCmfRYnoIUxcJOjCbeNi4QsIc/hhk+abxaltxFanESN+CNruhhOhs3JDpxs9NaE2/wDgZimTyVdsAMj+hWRwOUTtG4Mp8BACA1N5Aod6Hr1Lp2e/29KhSgULaSrnXlspVqKDNgDo1fDttD7qPU27hAgAheGDuF9tCk=" } }, { "type": "step", "result": "=\\int\\:\\sin^{5}\\left(u\\right)du" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\int\\:\\sin^{5}\\left(u\\right)du", "result": "=\\int\\:\\left(1-\\cos^{2}\\left(u\\right)\\right)^{2}\\sin\\left(u\\right)du", "steps": [ { "type": "interim", "title": "Simplify $$\\sin^{5}\\left(u\\right):{\\quad}\\sin^{4}\\left(u\\right)\\sin\\left(u\\right)$$", "input": "\\sin^{5}\\left(u\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "secondary": [ "$$\\sin^{5}\\left(u\\right)=\\sin^{4}\\left(u\\right)\\sin\\left(u\\right)$$" ], "result": "=\\sin^{4}\\left(u\\right)\\sin\\left(u\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7d/Ilys2F8HG6Q5muGWef7lXTSum/z5kLpMzXS1UJIewjFOcsU8TXhzIvoMfbQqJy8aXKAEgZ/lxWRK4baULsgmRLd2VwIqlBNByF6663syR2SpdpleAJc7YgKUwBYoM9Jx/Q9RJG0vE776DA5HR9xSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\int\\:\\sin^{4}\\left(u\\right)\\sin\\left(u\\right)du" }, { "type": "interim", "title": "Simplify $$\\sin^{4}\\left(u\\right):{\\quad}\\left(\\sin^{2}\\left(u\\right)\\right)^{2}$$", "input": "\\sin^{4}\\left(u\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$4$$ as $$2\\cdot\\:2$$", "result": "=\\sin^{2\\cdot\\:2}\\left(u\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c},\\:\\quad$$ assuming $$a\\ge0$$", "secondary": [ "$$\\sin^{2\\cdot\\:2}\\left(u\\right)=\\left(\\sin^{2}\\left(u\\right)\\right)^{2}$$" ], "result": "=\\left(\\sin^{2}\\left(u\\right)\\right)^{2}" } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SpLVILDk3Uy0+pZQrRAHlFXTSum/z5kLpMzXS1UJIeyBrgzvaHe1QwsRAKR9A2xmTInAIwJUYAOno3p/UxOCm0UqTd96MWTKI6Kr2Ib0iQAi5KYlQO0vFE/Inns2SruqFPM+1+uhKh8TZslNBT2/ow==" } }, { "type": "step", "result": "=\\int\\:\\left(\\sin^{2}\\left(u\\right)\\right)^{2}\\sin\\left(u\\right)du" }, { "type": "step", "primary": "Use the following identity: $$\\sin^{2}\\left(x\\right)=1-\\cos^{2}\\left(x\\right)$$", "result": "=\\int\\:\\left(1-\\cos^{2}\\left(u\\right)\\right)^{2}\\sin\\left(u\\right)du" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7655AYFq3t0ERjQdPukixI71fWutlCU4zUzgkFchUVlKOeWgsE4Mk40prEDZkKQ8Xcq64+b8YguXf4qCtKW9b7JFhxSzNcjgPzX10I7nnzbPguODNWb+2qEb4yZgx1CnXKmnfSPzRvFXP6y1kmr/cIUxK1tyiWP3evVB8/1lyBCeiD9kx/VS5qIswUMHVF3GBDMJZkL/6j5jtVUOIJSBOKqlxYnWxJTu1x/Pjz4hDH7Q" } }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\left(1-\\cos^{2}\\left(u\\right)\\right)^{2}\\sin\\left(u\\right)du", "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: $$v=\\cos\\left(u\\right)$$" ] }, { "type": "interim", "title": "$$\\frac{dv}{du}=-\\sin\\left(u\\right)$$", "input": "\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{du}\\left(\\cos\\left(u\\right)\\right)=-\\sin\\left(u\\right)$$", "result": "=-\\sin\\left(u\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuOCaaVOQ/y0RGnrcxaPJLr8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zBTby8v9dqkicCwl97RZujjd63ZtTkdUz4O+4dScYEckqbn5DUc+B2a9AsAORVtbh78yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=-\\sin\\left(u\\right)du$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\left(-\\frac{1}{\\sin\\left(u\\right)}\\right)dv$$" }, { "type": "step", "result": "=\\int\\:\\left(1-v^{2}\\right)^{2}\\sin\\left(u\\right)\\left(-\\frac{1}{\\sin\\left(u\\right)}\\right)dv" }, { "type": "interim", "title": "Simplify $$\\left(1-v^{2}\\right)^{2}\\sin\\left(u\\right)\\left(-\\frac{1}{\\sin\\left(u\\right)}\\right):{\\quad}-1+2v^{2}-v^{4}$$", "input": "\\left(1-v^{2}\\right)^{2}\\sin\\left(u\\right)\\left(-\\frac{1}{\\sin\\left(u\\right)}\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\left(1-v^{2}\\right)^{2}\\sin\\left(u\\right)\\frac{1}{\\sin\\left(u\\right)}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:\\left(1-v^{2}\\right)^{2}\\sin\\left(u\\right)}{\\sin\\left(u\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$\\sin\\left(u\\right)$$", "result": "=-1\\cdot\\:\\left(1-v^{2}\\right)^{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\left(1-v^{2}\\right)^{2}=\\left(1-v^{2}\\right)^{2}$$", "result": "=-\\left(-v^{2}+1\\right)^{2}" }, { "type": "interim", "title": "$$\\left(1-v^{2}\\right)^{2}:{\\quad}1-2v^{2}+v^{4}$$", "result": "=-\\left(1-2v^{2}+v^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=1,\\:\\:b=v^{2}$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=1^{2}-2\\cdot\\:1\\cdot\\:v^{2}+\\left(v^{2}\\right)^{2}" }, { "type": "interim", "title": "Simplify $$1^{2}-2\\cdot\\:1\\cdot\\:v^{2}+\\left(v^{2}\\right)^{2}:{\\quad}1-2v^{2}+v^{4}$$", "input": "1^{2}-2\\cdot\\:1\\cdot\\:v^{2}+\\left(v^{2}\\right)^{2}", "result": "=1-2v^{2}+v^{4}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=1-2\\cdot\\:1\\cdot\\:v^{2}+\\left(v^{2}\\right)^{2}" }, { "type": "interim", "title": "$$2\\cdot\\:1\\cdot\\:v^{2}=2v^{2}$$", "input": "2\\cdot\\:1\\cdot\\:v^{2}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2v^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFPqlt2vQGDCvjAbwsC8Hk2YJQJZuTAY5js+oqjdT8ksl8WAYsgXLn7jsPRjmRiUIcruiklADMVg+mqf/Zstq9e5kdUP3gKxLWQ8loibMCJvZZQwwh1vvJachWNAjqJifnw==" } }, { "type": "interim", "title": "$$\\left(v^{2}\\right)^{2}=v^{4}$$", "input": "\\left(v^{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=v^{2\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=v^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IPBtSEHcDmihEXEG+NuNo96GQqufR6tr2vPxOUv7H+9MqwtdIhEIfMoZUVsGLO7f8SrqrDW4mFcEK+hPNqZN8g5cW0YayS8zLPZlVTa5GIywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=1-2v^{2}+v^{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Expand $$-\\left(1-2v^{2}+v^{4}\\right):{\\quad}-1+2v^{2}-v^{4}$$", "result": "=-1+2v^{2}-v^{4}", "steps": [ { "type": "step", "primary": "Distribute parentheses", "result": "=-\\left(1\\right)-\\left(-2v^{2}\\right)-\\left(v^{4}\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-1+2v^{2}-v^{4}" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:-1+2v^{2}-v^{4}dv" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s75Y9KL8owIXwP9467uKBgmJssy3g+E9SCD2K4fXWokqZpN4cZPWgnwFqHQUcV4FHsfBLh5j/jJcd1Frv9s/1xSw0pWMfsJc1e/Z0+a/wFZqiAkJrxJ9WhTU3LgGWBFMt53XdN9tKtvo8HYKsmxAfUxjWwPs1+Gw97t4MeuaNjSYTj/Izzo1ADgwiEoxtxD1FLCiHHck1Nqm0241bazPM6oU=" } }, { "type": "step", "result": "=\\int\\:-1+2v^{2}-v^{4}dv" }, { "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": "=-\\int\\:1dv+\\int\\:2v^{2}dv-\\int\\:v^{4}dv" }, { "type": "interim", "title": "$$\\int\\:1dv=v$$", "input": "\\int\\:1dv", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:v" }, { "type": "step", "primary": "Simplify", "result": "=v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:2v^{2}dv=\\frac{2v^{3}}{3}$$", "input": "\\int\\:2v^{2}dv", "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": "=2\\cdot\\:\\int\\:v^{2}dv" }, { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:v^{2}dv", "result": "=2\\cdot\\:\\frac{v^{3}}{3}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{v^{2+1}}{2+1}" }, { "type": "interim", "title": "Simplify $$\\frac{v^{2+1}}{2+1}:{\\quad}\\frac{v^{3}}{3}$$", "input": "\\frac{v^{2+1}}{2+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=\\frac{v^{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{v^{3}}{3}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s72JH+pemvoXTixTOAjTY5TCo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7odgZ7xfXRIvvOzKegz31HuNgQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{2v^{3}}{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:v^{4}dv=\\frac{v^{5}}{5}$$", "input": "\\int\\:v^{4}dv", "steps": [ { "type": "interim", "title": "Apply the Power Rule", "input": "\\int\\:v^{4}dv", "result": "=\\frac{v^{5}}{5}", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$", "result": "=\\frac{v^{4+1}}{4+1}" }, { "type": "interim", "title": "Simplify $$\\frac{v^{4+1}}{4+1}:{\\quad}\\frac{v^{5}}{5}$$", "input": "\\frac{v^{4+1}}{4+1}", "steps": [ { "type": "step", "primary": "Add the numbers: $$4+1=5$$", "result": "=\\frac{v^{5}}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{v^{5}}{5}" } ], "meta": { "interimType": "Power Rule Top 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s77oAUID/u0VUcSpPS4BnLc2o/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7oep9qW+ZBvVnMbdlvtrns82gQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-v+\\frac{2v^{3}}{3}-\\frac{v^{5}}{5}" }, { "type": "interim", "title": "Substitute back", "input": "-v+\\frac{2v^{3}}{3}-\\frac{v^{5}}{5}", "result": "=-\\cos\\left(\\ln\\left(x\\right)\\right)+\\frac{2\\cos^{3}\\left(\\ln\\left(x\\right)\\right)}{3}-\\frac{\\cos^{5}\\left(\\ln\\left(x\\right)\\right)}{5}", "steps": [ { "type": "step", "primary": "Substitute back $$v=\\cos\\left(u\\right)$$", "result": "=-\\cos\\left(u\\right)+\\frac{2\\cos^{3}\\left(u\\right)}{3}-\\frac{\\cos^{5}\\left(u\\right)}{5}" }, { "type": "step", "primary": "Substitute back $$u=\\ln\\left(x\\right)$$", "result": "=-\\cos\\left(\\ln\\left(x\\right)\\right)+\\frac{2\\cos^{3}\\left(\\ln\\left(x\\right)\\right)}{3}-\\frac{\\cos^{5}\\left(\\ln\\left(x\\right)\\right)}{5}" } ], "meta": { "interimType": "Generic Substitute Back 0Eq" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=-\\cos\\left(\\ln\\left(x\\right)\\right)+\\frac{2\\cos^{3}\\left(\\ln\\left(x\\right)\\right)}{3}-\\frac{\\cos^{5}\\left(\\ln\\left(x\\right)\\right)}{5}+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=-\\cos(\\ln(x))+\\frac{2\\cos^{3}(\\ln(x))}{3}-\\frac{\\cos^{5}(\\ln(x))}{5}+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }