5.8-bo'lim : Aniq integrallarni almashtirish qoidasi
7. Iloji bo'lsa, quyidagi integralni baholang. Agar buning iloji bo'lmasa, nima uchun integralni baholash mumkin emasligini aniq tushuntiring.
\[ \int_{\pi }^{{\frac{{3\pi }}{2}}}{{6\sin \left( {2w} \right) - 7\cos \left( w \right)dw}}\]Yechim
Biz qilishimiz kerak bo'lgan birinchi qadam - almashtirishni amalga oshirish.
Shu nuqtada siz almashtirishlar bilan juda qulay bo'lishingiz kerak. Agar almashtirishlar sizga mos kelmasa, siz almashtirish bo'limlariga qaytib, u erda ba'zi muammolarni hal qilishingiz kerak.
O'zgartirishni o'rnatishdan oldin biz integralni buzishimiz kerak, chunki ikkinchi a'zo almashtirishga muhtoj emas. Shunday qilib,
\[ \int_{\pi }^{{\frac{{3\pi }}{2}}}{{6\sin \left( {2w} \right) - 7\cos \left( w \right)dw}} = \int_{\pi }^{{\frac{{3\pi }}{2}}}{{6\sin \left( {2w} \right)dw}} - \int_{\pi }^{{\frac{{3\pi }}{2}}}{{7\cos \left( w \right)dw}}\]Birinchi integralning o'rnini bosish shunday bo'ladi:
\[u = 2w\]2-qadam
Mana bu birinchi integral uchun haqiqiy almashtirish ishi.
\[\begin{align*}du & = 2dw\hspace{0.25in}\,\, \to \hspace{0.25in}dw = \frac{1}{2}du\\ w & = \pi :u = 2\pi \hspace{0.75in}w = \frac{{3\pi }}{2}:u = 3\pi \end{align*}\]Ushbu bo'lim uchun eslatmalarda qilganimizdek, biz integralni bajargandan so'ng orqaga almashtirish bilan shug'ullanmaslik uchun chegaralarni \(u\) ga aylantiramiz.
Bu erda almashtirishdan keyingi integral.
\[ \int_{\pi }^{{\frac{{3\pi }}{2}}}{{6\sin \left( {2w} \right) - 7\cos \left( w \right)dw}} = 3\int_{{2\pi }}^{{3\pi }}{{\sin \left( u \right)du}} - \int_{\pi }^{{\frac{{3\pi }}{2}}}{{7\cos \left( w \right)dw}}\]3-qadam
U holda integral,
\[\int_{\pi }^{{\frac{{3\pi }}{2}}}{{6\sin \left( {2w} \right) - 7\cos \left( w \right)dw}} = \left. { - 3\cos \left( u \right)} \right|_{2\pi }^{3\pi } - \left. {7\sin \left( w \right)} \right|_\pi ^{\frac{{3\pi }}{2}} = \left( {3 - \left( { - 3} \right)} \right) + \left( {7 - 0} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{13}}\]