7.2-bo'lim : Trig funksiyalarini o'z ichiga olgan integrallar
8. \( \displaystyle \int{{\cos \left( {3t} \right)\sin \left( {8t} \right)\,dt}}\) ni baholang.
Yechim
Bu muammo uchun haqiqatan ham unchalik ko'p narsa yo'q. Biz qilishimiz kerak bo'lgan narsa, sinus va kosinusning ko'paytmasini yig'indiga kamaytirish uchun ushbu bo'limda keltirilgan formuladan foydalanishdir. Shunday qilib,
\[\int{{\cos \left( {3t} \right)\sin \left( {8t} \right)\,dt}} = \int{{\frac{1}{2}\left[ {\sin \left( {8t - 3t} \right) + \sin \left( {8t + 3t} \right)} \right]\,dt}} = \frac{1}{2}\int{{\sin \left( {5t} \right) + \sin \left( {11t} \right)\,dt}}\]Formulaga e'tibor berganingizga ishonch hosil qiling! Ushbu bo'limda keltirilgan formulada kosinus o'rniga birinchi bo'lib sinus ko'rsatilgan. Formuladan to'g'ri foydalanganingizga ishonch hosil qiling!
2-qadam
Endi biz qilishimiz kerak bo'lgan narsa - bu integralni baholash.
\[\int{{\cos \left( {3t} \right)\sin \left( {8t} \right)\,dt}} = \frac{1}{2}\left( { - \frac{1}{5}\cos \left( {5t} \right) - \frac{1}{{11}}\cos \left( {11t} \right)} \right) + c = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{1}{{10}}\cos \left( {5t} \right) - \frac{1}{{22}}\cos \left( {11t} \right) + c}}\]