trigonometrik funksiyalar qatnashgan ifodalarni integrallash

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trigonometrik funksiyalar qatnashgan ifodalarni integrallash trigonometrik funksiyalar qatnashgan ifodalarni integrallash reja: 1. r( sinx, cosx) ko’rinishdagi funksiyalarni integrallash 2. r( sinnx, cosmx) ko’rinishdagi funksiyalarni integrallash 3. trigonometrik funksiyalar qatnashgan ifodalarni integrallash trigonometrik funksiyalar qatnashgan ifodalarni integrallash hamma trigonometrik funksiyalarni sinx va cosx orqali ratsional ko’rinishda ifodalash mumkin. bu ifodani r ( sinx,cosx) orqali belgilaymiz. endi r ( sinx,cosx) ko’rinishdagi ifodani integrallash kerak bo’lsin. bunday integralni belgilash yordamida z o’zgaruvchili ratsional funksiyaning integraliga almashtirish mumkin. integralni bunday almashtirish ratsionallashtirish deyiladi. haqiqatdan ham, desak, shuning uchun bunda r1(z)-z o’zgaruvchili ratsional funksiya. bunday almashtirish r(sinx,cosx) ko’rinishdagi har qanday funksiyani integrallashga imkon beradi, shuning uchun bunday almashtirish universal trigonometrik almashtirish deyiladi. lekin bunday almashtirish ko’pincha ancha murakkab ratsional funksiyaga olib keladi. shuning uchun, sodda o’rniga qo’yishlardan ham foydalansa bo’ladi. masalan: 1) agar r( sinx,cosx) funksiya sinx ga nisbatan toq bo’lsa, ya`ni r(-sinx,cosx)-r( sinx,cosx) bo’lsa, u holda z=cosx; dz=-sinxdx o’rniga qo’yish bu funksiyani ratsionallashtiradi. 2) …

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ol: integralni hisoblang. yechish: integral ostidagi funksiya sinx ga nisbatan toq funksiya . shuning uchun z=cosx; dz=-sinxdx( sinxdx=-dz almashtirishni bajaramiz: 4) agar r(sinx, cosx) funksiya sinx va cosx darajalarining ko’paytmasi bo’lsa, ya`ni ko’rinishdagi integralni hisoblash, m va n ga bog’liq holda turli o’rniga qo’yishlar bajariladi: a) agar n>0 va toq bo’lsa, u holda cosx=z; sinxdx=-dz o’rniga qo’yish integralni ratsionallashtiradi. b) agar m>0 va toq bo’lsa, u holda sinx=z; cosxdx=dz o’rniga qo’yish bajariladi. 4-misol: integralni hisoblang. yechish: cosx=z; sinxdx=-dz almashtirishni bajaramiz: v) agar ikkala n va m ko’rsatkichlar juft va nomanfiy bo’lsa, u holda trigonometriyadan ma`lum bo’lgan darajani pasaytirish formulalaridan foydalanamiz. 5-misol. integralni hisoblang. yechish: darajani pasaytirish formulasidan foydalanamiz. g) agar m+n=-2k(0 (juft, nomusbat) bo’lsa, u holda tgx=z yoki z=ctgx o’rniga qo’yish integralni darajali funksiyalarning integrallari yig’indisiga olib keladi. 6-misol. integralni hisoblang. yechish: bu yerda n=-3; m=-1; m+n=-4<0 7-misol . integralni hisoblang. yechish: bu yerda n=2, m=-6; n+m=-4<0 quyidagini almashtirishni bajaramiz. …

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- + = - - + = - + = - + + æ è ç ö ø ÷ = = - + + + = - + + + ò ò ò ò ò sin sin cos ( cos ) sin cos ( ) ln cos cos ln cos ; 2 2 2 2 2 2 2 1 2 1 2 1 2 2 3 2 2 2 3 2 2 2 3 2 sin cos n m x xdx × ò ò = dx x x j 4 3 cos sin ; cos 1 cos 3 1 1 3 1 ) 1 ( cos sin ) cos 1 ( cos sin sin 3 3 4 2 4 4 2 4 2 4 2 c x x c z z dz z z z dz z dz z x xdx x dx x x x j + - = …

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ò sin sin sin ln ln 3 2 2 2 2 2 2 3 2 2 2 2 3 2 4 3 3 2 2 2 2 1 2 1 2 1 2 1 1 4 2 1 4 1 1 4 1 2 1 4 1 2 1 8 1 2 1 4 2 1 8 2 1 2 2 1 8 2 2 x c + ; cos cos sin cos sin sin nx mxdx nx mxdx nx mxdx × × × ò ò ò ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] cos cos cos cos sin cos sin sin sin sin cos cos a b a b a b a b a b a b a b a b a b × = + + - × = + + - × = - - + …

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trigonometrik funksiyalar qatnashgan ifodalarni integrallash trigonometrik funksiyalar qatnashgan ifodalarni integrallash reja: 1. r( sinx, cosx) ko’rinishdagi funksiyalarni integrallash 2. r( sinnx, cosmx) ko’rinishdagi funksiyalarni integrallash 3. trigonometrik funksiyalar qatnashgan ifodalarni integrallash trigonometrik funksiyalar qatnashgan ifodalarni integrallash hamma trigonometrik funksiyalarni sinx va cosx orqali ratsional ko’rinishda ifodalash mumkin. bu ifodani r ( sinx,cosx) orqali belgilaymiz. endi r ( sinx,cosx) ko’rinishdagi ifodani integrallash kerak bo’lsin. bunday integralni belgilash yordamida z o’zgaruvchili ratsional funksiyaning integraliga almashtirish mumkin. integralni bunday almashtirish ratsionallashtirish deyiladi. haqiqatdan ham, desak, shuning uchun bunda r1(...

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