5.7-bo'lim : Aniq integrallarni hisoblash
1. Quyidagi integrallarning har birini baholang.
- \( \displaystyle \int{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\)
- \( \displaystyle \int_{{ - 3}}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\)
- \( \displaystyle \int_{1}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\)
a \( \displaystyle \int{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\) Yechim
Bu shunchaki noaniq integral va biz ularni bajarishda qulay bo'lishimiz kerak, shuning uchun bu qismga javob.
\[\int{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}} = \int{{\cos \left( x \right) - 3{x^{ - 5}}\,dx}} = \sin \left( x \right) + \frac{3}{4}{x^{ - 4}} + c = \require{bbox} \bbox[2pt,border:1px solid black]{{\sin \left( x \right) + \frac{3}{{4{x^4}}} + c}}\]"+c" ni qo'shishni unutmang, chunki biz noaniq integral qilamiz!
b \( \displaystyle \int_{{ - 3}}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\) Yechim
Eslatib o'tamiz, aniq integralni bajarish uchun (i.e. biz integratsiya qilayotgan funktsiya) biz integrallashayotgan intervalda uzluksiz bo'lishi kerak, bu holda \(\left[ { - 3,4} \right]\).
Biz aniq ko'ramizki, ikkinchi hadda \(x = 0\)da nolga bo'linish bo'ladi va \(x = 0\) biz integrallashayotgan intervalda va shuning uchun biz integrallashayotgan intervalda bu funksiya uzluksiz emas.
Shuning uchun bu integralni bajarib bo'lmaydi.
c \( \displaystyle \int_{1}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\) Yechim
Endi, funktsiya \(x = 0\) da ikkinchi muddatda nolga bo'linish muammosiga ega. Biroq, oldingi qismdan farqli o'laroq, \(x = 0\) biz birlashtirgan intervalga tushmaydi, bu holda \(\left[ {1,4} \right]\).
Shunday qilib, bu integral bajarilishi mumkin. Mana bu integral uchun ish.
\[\begin{align*}\int_{1}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}} & = \int_{1}^{4}{{\cos \left( x \right) - 3{x^{ - 5}}\,dx}} = \left. {\left( {\sin \left( x \right) + \frac{3}{{4{x^4}}}} \right)} \right|_1^4\\ & = \sin \left( 4 \right) + \frac{3}{{4\left( {{4^4}} \right)}} - \left( {\sin \left( 1 \right) + \frac{3}{{4\left( {{1^4}} \right)}}} \right)\\ & = \sin \left( 4 \right) + \frac{3}{{1024}} - \left( {\sin \left( 1 \right) + \frac{3}{4}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\sin \left( 4 \right) - \sin \left( 1 \right) - \frac{{765}}{{1024}}}}\end{align*}\]