5.3-bo'lim : Noaniq integrallarni almashtirish qoidasi
9. \( \displaystyle \int{{\frac{{4w + 3}}{{4{w^2} + 6w - 1}}\,dw}}\) ni baholang.
Maslahat: Maxrajning hosilasi nima?
Yechim
Bunday holda, biz o'rnini bosuvchi sifatida quyidagilarni ishlatishimiz kerak.
\[u = 4{w^2} + 6w - 1\]
Maslahat: Esda tutingki, almashtirishdan so'ng integraldagi barcha asl o'zgaruvchilar \(u\) bilan almashtirilishi kerak.
2-qadam
Barcha \(w\)lar \(u\) bilan almashtirilganligiga ishonch hosil qilishimiz kerak, biz differentsialni hisoblashimiz kerak, shunda biz \(dw\) va qolgan \(w\)larni integranddan olib tashlashimiz mumkin.
\[du = \left( {8w + 6} \right)dw\]O'zgartirishga yordam berish uchun keling, buni biroz qayta yozamiz,
\[du = 2\left( {4w + 3} \right)dw\,\,\hspace{0.25in} \Rightarrow \hspace{0.25in}\left( {4w + 3} \right)dw = \frac{1}{2}du\]3-qadam
O'zgartirishni amalga oshirish va integralni baholash,
\[\int{{\frac{{4w + 3}}{{4{w^2} + 6w - 1}}\,dw}} = \frac{1}{2}\int{{\frac{1}{u}\,du}} = \frac{1}{2}\ln \left| u \right| + c\]
Maslahat: Integranddagi asl o'zgaruvchi \(u\) emasligini unutmang!
4-qadam
Nihoyat, asl o'zgaruvchiga qaytishni unutmang!
\[\int{{\frac{{4w + 3}}{{4{w^2} + 6w - 1}}\,dw}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{2}\ln \left| {4{w^2} + 6w - 1} \right| + c}}\]