integral of 1/(cos(x)sin(x)+cos^2(x))

{ "query": { "display": "$$\\int\\:\\frac{1}{\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1}{\\cos(x)\\sin(x)+\\cos^{2}(x)}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "\\frac{1}{2}\\ln\\left|1-\\tan^{2}(x)\\right|+\\frac{1}{2}\\ln\\left|\\tan(2x)+\\sec(2x)\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1}{\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)}dx=\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(x\\right)\\right|+\\frac{1}{2}\\ln\\left|\\tan\\left(2x\\right)+\\sec\\left(2x\\right)\\right|+C$$", "input": "\\int\\:\\frac{1}{\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)}dx", "steps": [ { "type": "interim", "title": "Multiply by the conjugate of $$\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right):\\:\\frac{-\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}$$", "input": "\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)", "result": "=\\int\\:\\frac{1}{\\frac{-\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}}dx", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}$$", "result": "=\\frac{\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)\\right)\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}" }, { "type": "interim", "title": "Expand $$\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)\\right)\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)\\right):{\\quad}-\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)$$", "input": "\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)\\right)\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)\\right)", "steps": [ { "type": "interim", "title": "$$\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)=-\\cos\\left(2x\\right)$$", "input": "\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)", "steps": [ { "type": "interim", "title": "Expand $$\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right):{\\quad}\\sin^{2}\\left(x\\right)-\\cos^{2}\\left(x\\right)$$", "input": "\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)", "result": "=\\sin^{2}\\left(x\\right)-\\cos^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$\\left(a+b\\right)\\left(a-b\\right)=a^{2}-b^{2}$$", "secondary": [ "$$a=\\sin\\left(x\\right),\\:b=\\cos\\left(x\\right)$$" ], "result": "=\\sin^{2}\\left(x\\right)-\\cos^{2}\\left(x\\right)", "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Expand Difference of Squares" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cC2vx16a275Ce2/7HPHNaJpLRyyHUTF4keWRdUZ3nU8K8N1AvD/sjs+O3GJ0pZVBdYPfXQvX4/bINBB8wSEQ0TCZemGhnJ26dWCehvTfL5nHINhZ9+q4VDfZQA2foKJ7o3oe/oyhMy2+1TQhDBd2f4XZazvhtEMhdr5iLwjMPT6nrQ95xJxb0/iBFlfol3Uz4W0t+3k+sFrKUXIXwoCVSLsYMOfMWJINjcdqbrjG2Fk=" } }, { "type": "step", "primary": "Use the Double Angle identity: $$\\cos^{2}\\left(x\\right)-\\sin^{2}\\left(x\\right)=\\cos\\left(2x\\right)$$", "secondary": [ "$$-\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=-\\cos\\left(2x\\right)$$" ], "result": "=-\\cos\\left(2x\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\cos^{2}\\left(x\\right)\\left(-\\cos\\left(2x\\right)\\right)" }, { "type": "interim", "title": "Factor $$\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)\\right)\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)\\right):{\\quad}\\cos^{2}\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)$$", "input": "\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)\\right)\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)\\right)", "result": "=\\cos^{2}\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)", "steps": [ { "type": "interim", "title": "Factor $$\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right):{\\quad}\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)$$", "input": "\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos^{2}\\left(x\\right)", "result": "=\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)\\left(\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)=\\cos\\left(x\\right)\\cos\\left(x\\right)$$" ], "result": "=\\cos\\left(x\\right)\\sin\\left(x\\right)+\\cos\\left(x\\right)\\cos\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "interim", "title": "Factor $$\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right):{\\quad}\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)$$", "input": "\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)", "result": "=\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)=\\cos\\left(x\\right)\\cos\\left(x\\right)$$" ], "result": "=\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos\\left(x\\right)\\cos\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Refine", "result": "=\\cos^{2}\\left(x\\right)\\left(\\sin\\left(x\\right)+\\cos\\left(x\\right)\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=-\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s761C1I6HckjQ8wB9k8UNnejsXyd2ntXQlEAn8lwjgVzgpg2QlDXoBFDV8czBR0YeoMERZVLxN/Lg/eKiKCvsIFwCWKUbvV6WK3fDUgFtg3Q+Vu0y/nymFFvapFvN0D5+8cs7YyTnmUlHIXdyPqusBGvrIewIbgQd8+Nk4C+T0LHbe1A9ekahUIWtvNWrN4v94WJ7MAlyIlOhhjL1gSUtFjJp0GdLahlxcloLqWlCBNWSdwn6QklnfOj/iz73NvkbakFLb2Q7540VQhS6ZjLkSc7ZkYtBN3gtB3v8REqVC4VY=" } }, { "type": "step", "result": "=\\frac{-\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}" } ], "meta": { "interimType": "Multiply Divide Conjugate 2Eq" } }, { "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\\:\\frac{1}{\\frac{\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}}dx" }, { "type": "interim", "title": "Expand $$\\frac{1}{\\frac{\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}}:{\\quad}\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}-\\frac{1}{\\cos\\left(2x\\right)}$$", "input": "\\frac{1}{\\frac{\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)}{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{1}{\\frac{b}{c}}=\\frac{c}{b}$$", "result": "=\\frac{\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)}" }, { "type": "interim", "title": "Factor $$\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right):{\\quad}\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)$$", "input": "\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos^{2}\\left(x\\right)", "result": "=\\frac{\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)}{\\cos^{2}\\left(x\\right)\\cos\\left(2x\\right)}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)=\\cos\\left(x\\right)\\cos\\left(x\\right)$$" ], "result": "=\\cos\\left(x\\right)\\sin\\left(x\\right)-\\cos\\left(x\\right)\\cos\\left(x\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)-\\cos\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$\\cos\\left(x\\right)$$", "result": "=\\frac{\\sin\\left(x\\right)-\\cos\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{\\sin\\left(x\\right)-\\cos\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}-\\frac{\\cos\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}$$" ], "result": "=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}-\\frac{\\cos\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}" }, { "type": "interim", "title": "Cancel $$\\frac{\\cos\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}:{\\quad}\\frac{1}{\\cos\\left(2x\\right)}$$", "input": "\\frac{\\cos\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$\\cos\\left(x\\right)$$", "result": "=\\frac{1}{\\cos\\left(2x\\right)}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqY5ewE651e6m6D/ouNDQVC72U4SBMbmhmIphc/RtKoS3oZCq59Hq2va8/E5S/sf72Sa96ofM+5vPZoutcj6hmh92pSYNEMr9NFedeYhwYIb72wZm7kDUxdE6YSmfEbr2rA6UckXd5QhYMrrQC3e3Ib5PwvM7XKO3kL37Krbg4WN1ipklNwAYI3aqjqzKv7YArCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}-\\frac{1}{\\cos\\left(2x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nIyYUEjnwJO2hdBXNY/0co5XWCHPnyowIRNxqOR/NagycU28KAypNFvulkZhr333ofc85/oqy+yrbKtPyZ7PxUScXOS1iqwYhVyufRVfjd4DnzlbPZjyKgy1eUCFsLd59wUEHG2dTu2P2I+pIwlgwQVaH5VZwoTIHoZC5x/vXOo2f8rRAs4eOHhzsWZxtU+xhAoJoo+JbeTbUUsvSlAd7qN6Hv6MoTMtvtU0IQwXdn9PNSm+B7nW4/+8BgmEMMqr0Ynw7bx9xwgHh4VzF6hbxPPtUlbpXhZx4K8ZmqInRzOUNMAWK+azJYxoRUhclqtcELKzkfRLwhqzh627QQuMkbMukdvk8d8PpMtD+3DXp74=" } }, { "type": "step", "result": "=-\\int\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}-\\frac{1}{\\cos\\left(2x\\right)}dx" }, { "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\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}dx-\\int\\:\\frac{1}{\\cos\\left(2x\\right)}dx\\right)" }, { "type": "interim", "title": "$$\\int\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}dx=-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(x\\right)\\right|$$", "input": "\\int\\:\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)\\cos\\left(2x\\right)}dx", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\tan\\left(x\\right)$$", "result": "=\\int\\:\\frac{\\tan\\left(x\\right)}{\\cos\\left(2x\\right)}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\tan\\left(x\\right)}{\\cos\\left(2x\\right)}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: $$u=2x$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51iZ4v02Fm2dNqQJZnxCX4Je" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{2}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{\\tan\\left(x\\right)}{\\cos\\left(u\\right)}\\cdot\\:\\frac{1}{2}du" }, { "type": "interim", "title": "Simplify $$\\frac{\\tan\\left(x\\right)}{\\cos\\left(u\\right)}\\cdot\\:\\frac{1}{2}:{\\quad}\\frac{\\tan\\left(x\\right)}{2\\cos\\left(u\\right)}$$", "input": "\\frac{\\tan\\left(x\\right)}{\\cos\\left(u\\right)}\\cdot\\:\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{\\tan\\left(x\\right)\\cdot\\:1}{\\cos\\left(u\\right)\\cdot\\:2}" }, { "type": "step", "primary": "Multiply: $$\\tan\\left(x\\right)\\cdot\\:1=\\tan\\left(x\\right)$$", "result": "=\\frac{\\tan\\left(x\\right)}{2\\cos\\left(u\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{\\tan\\left(x\\right)}{2\\cos\\left(u\\right)}du" }, { "type": "interim", "title": "$$u=2x\\quad\\Rightarrow\\quad\\:x=\\frac{u}{2}$$", "input": "2x=u", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=u", "result": "x=\\frac{u}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{u}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{u}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "=\\int\\:\\frac{\\tan\\left(\\frac{u}{2}\\right)}{2\\cos\\left(u\\right)}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s71lewa+fm1gllISKpm3bbq0rZ3wSZocwjtUDIp+GK4jPSUM9pakkKILvT6Fs/PM35zdOFr14NhlnTuLeZXt0DnkvTqyge4+vpHW4VBaWAsV8Eoy0xTg26MoOt6jFtKKexmeVl3Sr6Lsj1TDP8q1VHm0/kIufF9OwaAuPjf2I6SDjZEt3ZXAiqUE0HIXrrrezJAIOLq+Zo6QPUdsELPTSSFYFODKn911wSXXfeQ1ST5CA" } }, { "type": "step", "result": "=\\int\\:\\frac{\\tan\\left(\\frac{u}{2}\\right)}{2\\cos\\left(u\\right)}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}{2}\\cdot\\:\\int\\:\\frac{\\tan\\left(\\frac{u}{2}\\right)}{\\cos\\left(u\\right)}du" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{\\tan\\left(\\frac{u}{2}\\right)}{\\cos\\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: $$v=\\tan\\left(\\frac{u}{2}\\right)$$" ] }, { "type": "step", "primary": "Using Weierstrass Substitution:" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:\\cos\\left(u\\right)=\\frac{1-v^{2}}{1+v^{2}},\\:\\tan\\left(\\frac{u}{2}\\right)=v$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=\\frac{2}{1+v^{2}}dv$$" }, { "type": "step", "result": "=\\int\\:\\frac{v}{\\frac{1-v^{2}}{1+v^{2}}}\\cdot\\:\\frac{2}{1+v^{2}}dv" }, { "type": "interim", "title": "Simplify $$\\frac{v}{\\frac{1-v^{2}}{1+v^{2}}}\\cdot\\:\\frac{2}{1+v^{2}}:{\\quad}\\frac{2v}{1-v^{2}}$$", "input": "\\frac{v}{\\frac{1-v^{2}}{1+v^{2}}}\\cdot\\:\\frac{2}{1+v^{2}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{v\\left(v^{2}+1\\right)}{-v^{2}+1}\\cdot\\:\\frac{2}{v^{2}+1}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{v\\left(1+v^{2}\\right)\\cdot\\:2}{\\left(1-v^{2}\\right)\\left(1+v^{2}\\right)}" }, { "type": "step", "primary": "Cancel the common factor: $$1+v^{2}$$", "result": "=\\frac{v\\cdot\\:2}{1-v^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{2v}{1-v^{2}}dv" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7xKTxOnyP2m1trLSF0ERscEQ5lfIU08cV7QYC/RfTanleL0LXjSZHEeyHU5TcRN7QVQS/Jt0lGNpQG/dCOLjzQekCgntDFdUQWCp8pL+1z0BPbm+EfPW2lf+ZDdfktozvWSzo9f74o6ikYmbvxINF4Us/9CZe3M0pPx3HG5wYcPiqpCF+pLHginRLYfnCC58Mu4jklVCgrx8HodpU+Hl0sPE2qmyRcAM78AN0YUwDzYC" } }, { "type": "step", "result": "=\\frac{1}{2}\\cdot\\:\\int\\:\\frac{2v}{1-v^{2}}dv" }, { "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}{2}\\cdot\\:2\\cdot\\:\\int\\:\\frac{v}{1-v^{2}}dv" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{v}{1-v^{2}}dv", "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: $$w=1-v^{2}$$" ] }, { "type": "interim", "title": "$$\\frac{dw}{dv}=-2v$$", "input": "\\frac{d}{dv}\\left(1-v^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dv}\\left(1\\right)-\\frac{d}{dv}\\left(v^{2}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(1\\right)=0$$", "input": "\\frac{d}{dv}\\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+idP5vLVrjUll6eMdYrYlw0sQkDeBfYmbOrQ+t91J8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtHFi1sZ1xQhHh3bCtbuI/B" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(v^{2}\\right)=2v$$", "input": "\\frac{d}{dv}\\left(v^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2v^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiHI6i/lNYJouYKcylLF6B2k3hxk9aCfAWodBRxXgUexsoRboyLdWbDdDvojbJb4SkeCBKuYKgaNJ253gLI69U5feCPJC8Uak4mwlsl/8zOjPWUEL+I3n8Z72JloyPMrWQ==" } }, { "type": "step", "result": "=0-2v" }, { "type": "step", "primary": "Simplify", "result": "=-2v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dw=-2vdv$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dv=\\left(-\\frac{1}{2v}\\right)dw$$" }, { "type": "step", "result": "=\\int\\:\\frac{v}{w}\\left(-\\frac{1}{2v}\\right)dw" }, { "type": "interim", "title": "Simplify $$\\frac{v}{w}\\left(-\\frac{1}{2v}\\right):{\\quad}-\\frac{1}{2w}$$", "input": "\\frac{v}{w}\\left(-\\frac{1}{2v}\\right)", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{v}{w}\\cdot\\:\\frac{1}{2v}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=-\\frac{v\\cdot\\:1}{w\\cdot\\:2v}" }, { "type": "step", "primary": "Cancel the common factor: $$v$$", "result": "=-\\frac{1}{w\\cdot\\:2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:-\\frac{1}{2w}dw" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+O3P4NyjST8qKRUS5Q9ARQBFOfoHFIkgHkATEbqyRla2NCwimM7dB8C524HvMCB4ymZOc9q9xxqJAg2jt99whbN2Iqnf8o89hIY+/5geo/Se523Vm7Rn9X/LRxylwEtyaN6Hv6MoTMtvtU0IQwXdn84k+SM9uK5ZgVIsXdnKy6Jugs8OYsrd0uK6mBpcEuDFg==" } }, { "type": "step", "result": "=\\frac{1}{2}\\cdot\\:2\\cdot\\:\\int\\:-\\frac{1}{2w}dw" }, { "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}{2}\\cdot\\:2\\left(-\\frac{1}{2}\\cdot\\:\\int\\:\\frac{1}{w}dw\\right)" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{w}dw=\\ln\\left(\\left|w\\right|\\right)$$", "result": "=\\frac{1}{2}\\cdot\\:2\\left(-\\frac{1}{2}\\ln\\left|w\\right|\\right)" }, { "type": "interim", "title": "Substitute back", "input": "\\frac{1}{2}\\cdot\\:2\\left(-\\frac{1}{2}\\ln\\left|w\\right|\\right)", "result": "=\\frac{1}{2}\\cdot\\:2\\left(-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(\\frac{2x}{2}\\right)\\right|\\right)", "steps": [ { "type": "step", "primary": "Substitute back $$w=1-v^{2}$$", "result": "=\\frac{1}{2}\\cdot\\:2\\left(-\\frac{1}{2}\\ln\\left|1-v^{2}\\right|\\right)" }, { "type": "step", "primary": "Substitute back $$v=\\tan\\left(\\frac{u}{2}\\right)$$", "result": "=\\frac{1}{2}\\cdot\\:2\\left(-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(\\frac{u}{2}\\right)\\right|\\right)" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=\\frac{1}{2}\\cdot\\:2\\left(-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(\\frac{2x}{2}\\right)\\right|\\right)" } ], "meta": { "interimType": "Generic Substitute Back 0Eq" } }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\cdot\\:2\\left(-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(\\frac{2x}{2}\\right)\\right|\\right):{\\quad}-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(x\\right)\\right|$$", "input": "\\frac{1}{2}\\cdot\\:2\\left(-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(\\frac{2x}{2}\\right)\\right|\\right)", "result": "=-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(x\\right)\\right|", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{2}\\cdot\\:2\\cdot\\:\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(\\frac{2x}{2}\\right)\\right|" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=-\\frac{1\\cdot\\:1\\cdot\\:2\\ln\\left|1-\\tan^{2}\\left(\\frac{2x}{2}\\right)\\right|}{2\\cdot\\:2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1\\cdot\\:1\\cdot\\:\\ln\\left|1-\\tan^{2}\\left(\\frac{2x}{2}\\right)\\right|}{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:1\\cdot\\:\\ln\\left|1-\\tan^{2}\\left(\\frac{2x}{2}\\right)\\right|=\\ln\\left|1-\\tan^{2}\\left(\\frac{2x}{2}\\right)\\right|$$", "result": "=-\\frac{\\ln\\left|-\\tan^{2}\\left(\\frac{2x}{2}\\right)+1\\right|}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=-\\frac{\\ln\\left|-\\tan^{2}\\left(x\\right)+1\\right|}{2}" }, { "type": "step", "result": "=-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(x\\right)\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8as9YPBpnhEx8+M2+qbvUgEjMEVAur7whRRLfiQvdeMwb54Xr02AtubQm1WeLZW3jKi79z0oL6jLjgREVentjdTzzn9AJ6W8l0HSf7pTBZeXebTKSUwQXHFFpEsrzHtUqZmOAMwQuoEaTkPPgcbx99VR3CiPD1AwAAAf/51EhnKagrf0F0/g8+ThHIHG53eOcUUqTd96MWTKI6Kr2Ib0iQAi5KYlQO0vFE/Inns2Sruq/PGQQt5/40znQVM0X/ZsoYalImQ9Ch8MZvMjIexObCTi9RmH3z7zKMlBqau75A5KuKwkVQM5ky8y5NzeAV2CQeF7h/lw5V186V1SutNm5g+wiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{1}{\\cos\\left(2x\\right)}dx=\\frac{1}{2}\\ln\\left|\\tan\\left(2x\\right)+\\sec\\left(2x\\right)\\right|$$", "input": "\\int\\:\\frac{1}{\\cos\\left(2x\\right)}dx", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\int\\:\\frac{1}{\\cos\\left(2x\\right)}dx", "result": "=\\int\\:\\sec\\left(2x\\right)dx", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\frac{1}{\\cos\\left(x\\right)}=\\sec\\left(x\\right)$$", "result": "=\\int\\:\\sec\\left(2x\\right)dx" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s74VUWXddkFRJ6FZazpSM+d5CFAOfj2uIfLcNIguBWPJjglS0lOYo+MEX6Bpkgk4kWW2OqI7u8uw/2TDZOKR8MlHIjRu93pF6Ud8rAuy8zIsm9eA3KzDR5LdoQ96EQOWijiBtScDPh1fpDokGg7yx6VK9dDZkXCs5+Cxg1YNvJNPUe9yYF2Q5qSv3gDlLQoBFjj5yGUAwgawpYTH/Vvtg3vE=" } }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\sec\\left(2x\\right)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: $$u=2x$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=2$$", "input": "\\frac{d}{dx}\\left(2x\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=2\\frac{dx}{dx}" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=2\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=2", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51iZ4v02Fm2dNqQJZnxCX4Je" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2dx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{2}du$$" }, { "type": "step", "result": "=\\int\\:\\sec\\left(u\\right)\\frac{1}{2}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+kjIfqCK8QGwzUQY8+wMOQsjvX7KVUO/AeCFSId4S33cZqb2ujmN2FEZC5M/msYIHiX35dQ/h01lIvxamZtt5PM+j+1P04cisiXW8HcDKOk4Ruz4fIbA7DSu1jg0w5EAYEFMST8lDZxn1Yq5HMKVTsGLeMxVl55xMYxEfjKBatulcQUlMOhkqQvF9O8Q8/Z5g==" } }, { "type": "step", "result": "=\\int\\:\\sec\\left(u\\right)\\frac{1}{2}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}{2}\\cdot\\:\\int\\:\\sec\\left(u\\right)du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\sec\\left(u\\right)du=\\ln\\left|\\tan\\left(u\\right)+\\sec\\left(u\\right)\\right|$$", "result": "=\\frac{1}{2}\\ln\\left|\\tan\\left(u\\right)+\\sec\\left(u\\right)\\right|" }, { "type": "step", "primary": "Substitute back $$u=2x$$", "result": "=\\frac{1}{2}\\ln\\left|\\tan\\left(2x\\right)+\\sec\\left(2x\\right)\\right|" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=-\\left(-\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(x\\right)\\right|-\\frac{1}{2}\\ln\\left|\\tan\\left(2x\\right)+\\sec\\left(2x\\right)\\right|\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(x\\right)\\right|+\\frac{1}{2}\\ln\\left|\\tan\\left(2x\\right)+\\sec\\left(2x\\right)\\right|", "meta": { "solvingClass": "Solver" } }, { "type": "step", "primary": "Add a constant to the solution", "result": "=\\frac{1}{2}\\ln\\left|1-\\tan^{2}\\left(x\\right)\\right|+\\frac{1}{2}\\ln\\left|\\tan\\left(2x\\right)+\\sec\\left(2x\\right)\\right|+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=\\frac{1}{2}\\ln\\left|1-\\tan^{2}(x)\\right|+\\frac{1}{2}\\ln\\left|\\tan(2x)+\\sec(2x)\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }