5.4-bo'lim : Ko'proq almashtirish qoidasi
3. \( \displaystyle \int{{\frac{{6{{\bf{e}}^{7w}}}}{{{{\left( {1 - 8{{\bf{e}}^{7w}}} \right)}^3}}} + \frac{{14{{\bf{e}}^{7w}}}}{{1 - 8{{\bf{e}}^{7w}}}}\,dw}}\) ni baholang.
Yechim
Ushbu integranda ikkita atama borligidan juda hayajonlanmang. Har bir atama bir xil almashtirishni talab qiladi,
\[u = 1 - 8{{\bf{e}}^{7w}}\]shuning uchun biz buni ikkala atamada ham ishlatamiz.
Agar siz asosiy almashtirish mexanikasidan mamnun bo'lmasangiz, oldingi bo'limda ba'zi muammolar bilan shug'ullanishingiz kerak, chunki biz ushbu bo'limda asoslar haqida batafsil ma'lumot bermaymiz. Ushbu bo'limdagi muammolar almashtirishning asosiy mexanikasi bilan ancha qulay bo'lganlar uchun mo'ljallangan va bir nechta "ilg'or" almashtirishlarni o'z ichiga oladi.
2-qadam
Bu erda almashtirish uchun differentsial ish.
\[du = - 56{{\bf{e}}^{7w}}dw\hspace{0.25in} \to \hspace{0.25in}\,{{\bf{e}}^{7w}}dw = - \frac{1}{{56}}du\]Haqiqiy almashtirishni amalga oshirishdan oldin \({{\bf{e}}^{7w}}\) ni integraldan quyidagi tarzda ajratib ko'rsatish qulay bo'lishi mumkin.
\[\int{{\frac{{6{{\bf{e}}^{7w}}}}{{{{\left( {1 - 8{{\bf{e}}^{7w}}} \right)}^3}}} + \frac{{14{{\bf{e}}^{7w}}}}{{1 - 8{{\bf{e}}^{7w}}}}\,dw}} = \int{{\left[ {\frac{6}{{{{\left( {1 - 8{{\bf{e}}^{7w}}} \right)}^3}}} + \frac{{14}}{{1 - 8{{\bf{e}}^{7w}}}}} \right]{{\bf{e}}^{7w}}\,dw}}\]Buni qilish differensial qismni yaratishi kerak (i.e. almashtirishning \(du\) qismi) aniqroq.
Endi almashtirishni amalga oshirish va integralni baholash,
\[\begin{align*}\int{{\frac{{6{{\bf{e}}^{7w}}}}{{{{\left( {1 - 8{{\bf{e}}^{7w}}} \right)}^3}}} + \frac{{14{{\bf{e}}^{7w}}}}{{1 - 8{{\bf{e}}^{7w}}}}\,dw}} & = - \frac{1}{{56}}\int{{6{u^{ - 3}} + \frac{{14}}{u}\,du}} = - \frac{1}{{56}}\left( { - 3{u^{ - 2}} + 14\ln \left| u \right|} \right) + c\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{1}{{56}}\left( { - 3{{\left( {1 - 8{{\bf{e}}^{7w}}} \right)}^{ - 2}} + 14\ln \left| {1 - 8{{\bf{e}}^{7w}}} \right|} \right) + c}}\end{align*}\]Bu yerda \(dw\) almashtirish bilan shug'ullanayotganda ehtiyot bo'ling. \( - \frac{1}{{56}}\) atamalardan biriga emas, balki butun integrandga ko‘paytirilishiga ishonch hosil qiling. Buni butun integrand atrofidagi qavslar yordamida yoki \( - \frac{1}{{56}}\) ni integraldan butunlay chiqarib tashlashingiz mumkin (bu yerda qilganimiz kabi).
Integralni baholagandan so'ng asl o'zgaruvchiga qaytishni unutmang!