7.2-bo'lim : Trig funksiyalarini o'z ichiga olgan integrallar
11. \( \displaystyle \int{{{{\csc }^6}\left( {\frac{1}{4}w} \right){{\cot }^4}\left( {\frac{1}{4}w} \right)\,dw}}\) ni baholang.
Yechim
Haqiqiy funktsiyalardagi aniq farqdan tashqari, bu muammo va tangens va sekantlarga ega bo'lgan muammo qanday ishlashida amaliy farq yo'q. Shunday qilib, biz qilishimiz kerak bo'lgan yagona narsa, agar u tangentlar va sekantlarni o'z ichiga olgan bo'lsa, bu qanday ishlaydi va biz bu borada ham ishlay olamiz.
Bu erda biz birinchi navbatda shuni ko'rishimiz mumkinki, kosekantdagi ko'rsatkich teng va shuning uchun biz ulardan ikkitasini ajratib olishimiz mumkin.
\[ \int{{{{\csc }^6}\left( {\frac{1}{4}w} \right){{\cot }^4}\left( {\frac{1}{4}w} \right)\,dw}} = \int{{{{\csc }^4}\left( {\frac{1}{4}w} \right){{\cot }^4}\left( {\frac{1}{4}w} \right)\,\,\,{{\csc }^2}\left( {\frac{1}{4}w} \right)\,dw}}\]2-qadam
Endi qolgan kosekantlarni kotangentlarga aylantirish uchun \({\cot ^2}\theta + 1 = {\csc ^2}\theta \) trig identifikatoridan foydalanishimiz mumkin.
\[\begin{align*}\int{{{{\csc }^6}\left( {\frac{1}{4}w} \right){{\cot }^4}\left( {\frac{1}{4}w} \right)\,dw}} & = \int{{{{\left[ {{{\csc }^2}\left( {\frac{1}{4}w} \right)} \right]}^2}{{\cot }^4}\left( {\frac{1}{4}w} \right)\,\,\,{{\csc }^2}\left( {\frac{1}{4}w} \right)\,dw}}\\ & = \int{{{{\left[ {{{\cot }^2}\left( {\frac{1}{4}w} \right) + 1} \right]}^2}{{\cot }^4}\left( {\frac{1}{4}w} \right)\,\,\,{{\csc }^2}\left( {\frac{1}{4}w} \right)\,dw}}\end{align*}\]3-qadam
Endi biz integralni baholash uchun \(u = \cot \left( {\frac{1}{4}w} \right)\) almashtirishdan foydalanishimiz mumkin.
\[\begin{align*}\int{{{{\csc }^6}\left( {\frac{1}{4}w} \right){{\cot }^4}\left( {\frac{1}{4}w} \right)\,dw}} & = - 4\int{{{{\left[ {{u^2} + 1} \right]}^2}{u^4}\,du}}\\ & = - 4\int{{{u^8} + 2{u^6} + {u^4}\,du}} = - 4\left( {\frac{1}{9}{u^9} + \frac{2}{7}{u^7} + \frac{1}{5}{u^5}} \right) + c\end{align*}\]E'tibor bering, biz bu erda haqiqiy almashtirish ishlarini qilmaymiz. Shu nuqtada, agar kerak bo'lsa, tafsilotlarni to'ldirish uchun almashtirishni yaxshi eslaysiz deb taxmin qilinadi. Agar siz almashtirishda zanglagan bo'lsangiz, ehtimol hisob-kitoblar bo'yicha muammolarga qaytishingiz va almashtirishlar bo'yicha mashq qilishingiz kerak.
4-qadam
\(u\) o'rniga qayta kirishni unutmang!
\[\int{{{{\csc }^6}\left( {\frac{1}{4}w} \right){{\cot }^4}\left( {\frac{1}{4}w} \right)\,dw}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{4}{9}{{\cot }^9}\left( {\frac{1}{4}w} \right) - \frac{8}{7}{{\cot }^7}\left( {\frac{1}{4}w} \right) - \frac{4}{5}{{\cot }^5}\left( {\frac{1}{4}w} \right) + c}}\]