7.4-bo'lim : qisman kasrlar
5. \( \displaystyle \int_{2}^{4}{{\frac{{3{z^2} + 1}}{{\left( {z + 1} \right){{\left( {z - 5} \right)}^2}}}\,dz}}\) integralini baholang.
Yechim
Bunday holda, maxraj allaqachon faktorlarga ajratilgan va shuning uchun biz to'g'ridan-to'g'ri integratsiya uchun qisman kasr parchalanish shakliga o'tishimiz mumkin.
\[\frac{{3{z^2} + 1}}{{\left( {z + 1} \right){{\left( {z - 5} \right)}^2}}} = \frac{A}{{z + 1}} + \frac{B}{{z - 5}} + \frac{C}{{{{\left( {z - 5} \right)}^2}}}\]2-qadam
Numeratorlarni teng belgilash,
\[3{z^2} + 1 = A{\left( {z - 5} \right)^2} + B\left( {z + 1} \right)\left( {z - 5} \right) + C\left( {z + 1} \right)\]3-qadam
Biz ikkita koeffitsientni osongina olish uchun eslatmalarda muhokama qilingan "hiyla" dan foydalanishimiz mumkin va keyin uchinchisini olish uchun \(z\) ning boshqa qiymatini tanlashimiz mumkin, shuning uchun buni qilaylik. Mana o'sha ish.
\[\begin{align*}z = & - 1: & 4 = & \,36A\\ & & & \\ z = & \,5 \,\,\,\,\,\, : & 76 = & \,6C\\ & & & \\z = & \,0 \,\,\,\,\,\, : & 1 = & \,25A - 5B + C = \frac{{139}}{9} - 5B\end{align*}\hspace{0.25in} \Rightarrow \hspace{0.25in}\,\begin{aligned} & {A = \frac{1}{9}}\\& {B = \frac{{26}}{9}}\\ & {C = \frac{{38}}{3}}\end{aligned}\]Integratsiyaning qisman kasr shakli shunday bo'ladi:
\[\frac{{3{z^2} + 1}}{{\left( {z + 1} \right){{\left( {z - 5} \right)}^2}}} = \frac{{\frac{1}{9}}}{{z + 1}} + \frac{{\frac{{26}}{9}}}{{z - 5}} + \frac{{\frac{{38}}{3}}}{{{{\left( {z - 5} \right)}^2}}}\]4-qadam
Endi biz integralni bajarishimiz mumkin.
\[\begin{align*}\int_{2}^{4}{{\frac{{3{z^2} + 1}}{{\left( {z + 1} \right){{\left( {z - 5} \right)}^2}}}\,dz}} & = \int_{2}^{4}{{\frac{{\frac{1}{9}}}{{z + 1}} + \frac{{\frac{{26}}{9}}}{{z - 5}} + \frac{{\frac{{38}}{3}}}{{{{\left( {z - 5} \right)}^2}}}\,dz}}\\ & = \left. {\left( {\frac{1}{9}\ln \left| {z + 1} \right| + \frac{{26}}{9}\ln \left| {z - 5} \right| - \frac{{\frac{{38}}{3}}}{{z - 5}}} \right)} \right|_2^4 = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{9}\ln \left( 5 \right) - \frac{{27}}{9}\ln \left( 3 \right) + \frac{{76}}{9}}}\end{align*}\]