Endi qolgan kosinuslarni sinuslarga aylantirish uchun \({\sin ^2}\theta + {\cos ^2}\theta = 1\) trig identifikatoridan foydalanishimiz mumkin.

\[\begin{align*}\int{{{{\sin }^8}\left( {3z} \right){{\cos }^5}\left( {3z} \right)\,dz}} & = \int{{{{\sin }^8}\left( {3z} \right){{\left[ {{{\cos }^2}\left( {3z} \right)} \right]}^2}\cos \left( {3z} \right)\,dz}}\\ & = \int{{{{\sin }^8}\left( {3z} \right){{\left[ {1 - {{\sin }^2}\left( {3z} \right)} \right]}^2}\cos \left( {3z} \right)\,dz}}\end{align*}\]

Endi biz integralni baholash uchun \(u = \sin \left( {3z} \right)\) almashtirishdan foydalanishimiz mumkin.

\[\begin{align*}\int{{{{\sin }^8}\left( {3z} \right){{\cos }^5}\left( {3z} \right)\,dz}} & = \frac{1}{3}\int{{{u^8}{{\left[ {1 - {u^2}} \right]}^2}\,du}}\\ & = \frac{1}{3}\int{{{u^8} - 2{u^{10}} + {u^{12}}\,du = \frac{1}{3}\left( {\frac{1}{9}{u^9} - \frac{2}{{11}}{u^{11}} + \frac{1}{{13}}{u^{13}}} \right) + c}}\end{align*}\]

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.

\(u\) o'rniga qayta kirishni unutmang!

\[ \int{{{{\sin }^8}\left( {3z} \right){{\cos }^5}\left( {3z} \right)\,dz}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{{27}}{{\sin }^9}\left( {3z} \right) - \frac{2}{{33}}{{\sin }^{11}}\left( {3z} \right) + \frac{1}{{39}}{{\sin }^{13}}\left( {3z} \right) + c}}\]