7.4-bo'lim : qisman kasrlar
1. \( \displaystyle \int{{\frac{4}{{{x^2} + 5x - 14}}\,dx}}\) integralini baholang.
Yechim
Muammoni boshlash uchun bizga integralning qisman kasr parchalanishi shakli kerak. Biroq, buni olish uchun biz maxrajni hisobga olishimiz kerak.
\[\int{{\frac{4}{{{x^2} + 5x - 14}}\,dx}} = \int{{\frac{4}{{\left( {x + 7} \right)\left( {x - 2} \right)}}\,dx}}\]U holda integrand uchun qisman kasrning parchalanish shakli quyidagicha:
\[\frac{4}{{\left( {x + 7} \right)\left( {x - 2} \right)}} = \frac{A}{{x + 7}} + \frac{B}{{x - 2}}\]2-qadam
Numeratorlarni teng belgilash,
\[4 = A\left( {x - 2} \right) + B\left( {x + 7} \right)\]3-qadam
Biz bu holatda koeffitsientlarni osongina olish uchun eslatmalarda muhokama qilingan "hiyla" dan foydalanishimiz mumkin, shuning uchun buni qilaylik. Mana o'sha ish.
\[\begin{align*}x = & 2 \,\,\,\,\,\,\,:& 4 & = 9B\\ & & & \\ x = & - 7:&4 & = - 9A\end{align*}\hspace{0.25in} \Rightarrow \hspace{0.25in}\begin{aligned} & {A = - \frac{4}{9}}\\ &{B = \frac{4}{9}}\end{aligned}\]Integratsiyaning qisman kasr shakli shunday bo'ladi:
\[\frac{4}{{\left( {x + 7} \right)\left( {x - 2} \right)}} = \frac{{ - \frac{4}{9}}}{{x + 7}} + \frac{{\frac{4}{9}}}{{x - 2}}\]4-qadam
Endi biz integralni bajarishimiz mumkin.
\[\int{{\frac{4}{{\left( {x + 7} \right)\left( {x - 2} \right)}}\,dx}} = \int{{\frac{{ - \frac{4}{9}}}{{x + 7}} + \frac{{\frac{4}{9}}}{{x - 2}}\,dx}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{4}{9}\ln \left| {x - 2} \right| - \frac{4}{9}\ln \left| {x + 7} \right| + c}}\]