7.4-bo'lim : qisman kasrlar
2. \( \displaystyle \int{{\frac{{8 - 3t}}{{10{t^2} + 13t - 3}}\,dt}}\) integralini baholang.
Yechim
Muammoni boshlash uchun bizga integralning qisman kasr parchalanishi shakli kerak. Biroq, buni olish uchun biz maxrajni hisobga olishimiz kerak.
\[\int{{\frac{{8 - 3t}}{{10{t^2} + 13t - 3}}\,dt}} = \int{{\frac{{8 - 3t}}{{\left( {2t + 3} \right)\left( {5t - 1} \right)}}\,dt}}\]U holda integrand uchun qisman kasrning parchalanish shakli quyidagicha:
\[\frac{{8 - 3t}}{{10{t^2} + 13t - 3}} = \frac{A}{{2t + 3}} + \frac{B}{{5t - 1}}\]2-qadam
Numeratorlarni teng belgilash,
\[8 - 3t = A\left( {5t - 1} \right) + B\left( {2t + 3} \right)\]3-qadam
Biz bu holatda koeffitsientlarni osongina olish uchun eslatmalarda muhokama qilingan "hiyla" dan foydalanishimiz mumkin, shuning uchun buni qilaylik. Mana o'sha ish.
\[\begin{align*}t = & \frac{1}{5} \,\,\,\,\,\,\, : & \frac{{37}}{5} & = \frac{{17}}{5}B\\ & & & \\t = & - \frac{3}{2}: & \frac{{25}}{2} & = - \frac{{17}}{2}A\end{align*}\hspace{0.25in} \Rightarrow \hspace{0.25in}\begin{aligned}&{A = - \frac{{25}}{{17}}}\\ & \\&{B = \frac{{37}}{{17}}}\end{aligned}\]Integratsiyaning qisman kasr shakli shunday bo'ladi:
\[\frac{{8 - 3t}}{{10{t^2} + 13t - 3}} = \frac{{ - \frac{{25}}{{17}}}}{{2t + 3}} + \frac{{\frac{{37}}{{17}}}}{{5t - 1}}\]4-qadam
Endi biz integralni bajarishimiz mumkin.
\[\int{{\frac{{8 - 3t}}{{10{t^2} + 13t - 3}}}}\,dt = \int{{\frac{{ - \frac{{25}}{{17}}}}{{2t + 3}} + \frac{{\frac{{37}}{{17}}}}{{5t - 1}}\,dt}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{37}}{{85}}\ln \left| {5t - 1} \right| - \frac{{25}}{{34}}\ln \left| {2t + 3} \right| + c}}\]Umid qilamanki, siz integratsiya bilan etarlicha yaxshi ishlayapsiz, bu integrallarning ba'zilarini boshingizda bajarishingiz mumkin. Biroq, bu ikkala integral bilan ehtiyot bo'ling. Bizning boshimizda bunday integrallarni bajarayotganda, texnik jihatdan ularni bajarish uchun zarur bo'lgan almashtirishlarni unutish va javobda ko'rsatilishi kerak bo'lgan almashtirish koeffitsientlarini o'tkazib yuborish oson.