Haqiqatan ham bu muammo uchun juda ko'p narsa yo'q. Eslatib o'tamiz, eslatmalardagi formula \(x = f\left( t \right)\) va \(y = g\left( t \right)\) ni nazarda tutadi. Demak, egri chiziq ostidagi maydon,

\[\begin{align*}A & = \int_{0}^{\pi }{{\left( {4 + \sin \left( t \right)} \right)\left( {3{{\cos }^2}\left( t \right)\sin \left( t \right)} \right)\,dt}}\\ & = \int_{0}^{\pi }{{12{{\cos }^2}\left( t \right)\sin \left( t \right) + 3{{\cos }^2}\left( t \right){{\sin }^2}\left( t \right)\,dt}}\\ & = \int_{0}^{\pi }{{12{{\cos }^2}\left( t \right)\sin \left( t \right) + 3{{\left[ {\frac{1}{2}\sin \left( {2t} \right)} \right]}^2}\,dt}}\\ & = \int_{0}^{\pi }{{12{{\cos }^2}\left( t \right)\sin \left( t \right) + \frac{3}{4}{{\sin }^2}\left( {2t} \right)\,dt}}\\ & = \int_{0}^{\pi }{{12{{\cos }^2}\left( t \right)\sin \left( t \right) + \frac{3}{8}\left( {1 - cos\left( {4t} \right)} \right)\,dt}}\\ & = \left. {\left( { - 4{{\cos }^3}\left( t \right) + \frac{3}{8}t - \frac{3}{{32}}\sin \left( {4t} \right)} \right)} \right|_{0}^{\pi } = \require{bbox} \bbox[2pt,border:1px solid black]{{8 + \frac{3}{8}\pi }}\end{align*}\]

Ushbu integralni to'g'ri bajarish uchun barcha trig manipulyatsiyalari va trig integrallarini qanday qilish kerakligini esladingizmi? Agar yo'q bo'lsa, siz unga qaytishingiz kerak Trig funksiyalarni o'z ichiga olgan integrallar Ba'zi ko'rib chiqish/muammolarni bajarish uchun bo'lim.