5.8-bo'lim : Aniq integrallarni almashtirish qoidasi
1. Iloji bo'lsa, quyidagi integralni baholang. Agar buning iloji bo'lmasa, nima uchun integralni baholash mumkin emasligini aniq tushuntiring.
\[ \int_{0}^{1}{{3\left( {4x + {x^4}} \right){{\left( {10{x^2} + {x^5} - 2} \right)}^6}\,dx}}\]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.
Ushbu muammoning o'rnini bosish,
\[u = 10{x^2} + {x^5} - 2\]2-qadam
Mana bu muammo uchun haqiqiy almashtirish ishi.
\[\begin{align*} du & = \left( {20x + 5{x^4}} \right)dx = 5\left( {4x + {x^4}} \right)dx\hspace{0.25in} \to \hspace{0.25in}\left( {4x + {x^4}} \right)dx = \frac{1}{5}du\\ x & = 0:u = - 2\hspace{0.75in}x = 1:u = 9\end{align*}\]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.
\[ \int_{0}^{1}{{3\left( {4x + {x^4}} \right){{\left( {10{x^2} + {x^5} - 2} \right)}^6}\,dx}} = \frac{3}{5}\int_{{ - 2}}^{9}{{{u^6}\,du}}\]3-qadam
U holda integral,
\[ \int_{0}^{1}{{3\left( {4x + {x^4}} \right){{\left( {10{x^2} + {x^5} - 2} \right)}^6}\,dx}} = \left. {\frac{3}{{35}}{u^7}} \right|_{ - 2}^9 = \frac{3}{{35}}\left( {4,782,969 - \left( { - 128} \right)} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{14,349,291}}{{35}}}}\]"Charqora" yoki "katta" javoblardan hayajonlanmang. Ular vaqti-vaqti bilan sodir bo'ladi, shuning uchun sodir bo'lganda ular haqida tashvishlanmang.