7.4-bo'lim : qisman kasrlar
3. \( \displaystyle \int_{{ - 1}}^{0}{{\frac{{{w^2} + 7w}}{{\left( {w + 2} \right)\left( {w - 1} \right)\left( {w - 4} \right)}}\,dw}}\) 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.
\[\frac{{{w^2} + 7w}}{{\left( {w + 2} \right)\left( {w - 1} \right)\left( {w - 4} \right)}} = \frac{A}{{w + 2}} + \frac{B}{{w - 1}} + \frac{C}{{w - 4}}\]2-qadam
Numeratorlarni teng belgilash,
\[{w^2} + 7w = A\left( {w - 1} \right)\left( {w - 4} \right) + B\left( {w + 2} \right)\left( {w - 4} \right) + C\left( {w + 2} \right)\left( {w - 1} \right)\]3-qadam
Biz bu holatda koeffitsientlarni osongina olish uchun eslatmalarda muhokama qilingan "hiyla" dan foydalanishimiz mumkin, shuning uchun buni qilaylik. Mana o'sha ish.
\[\begin{align*}w = & \; 1 \,\,\,\,\,\, : & 8 & = - 9B\\ & & & \\ w = & \, 4 \,\,\,\,\,\, : & 44 & = 18C\\ & & & \\ w = & - 2: & - 10 & = 18A\end{align*}\hspace{0.25in} \Rightarrow \hspace{0.25in}\begin{aligned}& {A = - \frac{5}{9}}\\ & {B = - \frac{8}{9}}\\ & {C = \frac{{22}}{9}}\end{aligned}\]Integratsiyaning qisman kasr shakli shunday bo'ladi:
\[\frac{{{w^2} + 7w}}{{\left( {w + 2} \right)\left( {w - 1} \right)\left( {w - 4} \right)}} = \frac{{ - \frac{5}{9}}}{{w + 2}} - \frac{{\frac{8}{9}}}{{w - 1}} + \frac{{\frac{{22}}{9}}}{{w - 4}}\]4-qadam
Endi biz integralni bajarishimiz mumkin.
\[\begin{align*}\int_{{ - 1}}^{0}{{\frac{{{w^2} + 7w}}{{\left( {w + 2} \right)\left( {w - 1} \right)\left( {w - 4} \right)}}\,dw}} & = \int_{{ - 1}}^{0}{{\frac{{ - \frac{5}{9}}}{{w + 2}} - \frac{{\frac{8}{9}}}{{w - 1}} + \frac{{\frac{{22}}{9}}}{{w - 4}}\,dw}}\\ & = \left. {\left( { - \frac{5}{9}\ln \left| {w + 2} \right| - \frac{8}{9}\ln \left| {w - 1} \right| + \frac{{22}}{9}\ln \left| {w - 4} \right|} \right)} \right|_{ - 1}^0\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{22}}{9}\ln \left( 4 \right) + \frac{3}{9}\ln \left( 2 \right) - \frac{{22}}{9}\ln \left( 5 \right) = \frac{{47}}{9}\ln \left( 2 \right) - \frac{{22}}{9}\ln \left( 5 \right)}}\end{align*}\]E'tibor bering, biz birinchi ikkita logarifmni bitta logarifmga birlashtirish uchun tezkor logarifm xususiyatidan foydalanganmiz. Agar biz ishlatgan logarifmni tanmasangiz, ehtimol siz o'zingizning logarifm xususiyatlarini ko'rib chiqishingiz kerak. Bunday mulk ilovalari, agar siz ularni bilsangiz, ba'zida ishingizni chindan ham soddalashtirishi mumkin!