∫ (from 0 to 3) of (135+95x)(6+x) wrt x eching

$$\int_{ 0 }^{ 3 } (135+95x)(6+x) d x$$

Baholash

$$\frac{12915}{2}=6457,5$$

Yechim qadamlari

135+95x ifodaning har bir elementini 6+x ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$\int _{0}^{3}810+135x+570x+95x^{2}\mathrm{d}x$$

705x ni olish uchun 135x va 570x ni birlashtirish.

$$\int _{0}^{3}810+705x+95x^{2}\mathrm{d}x$$

Avval noaniq integralni baholang.

$$\int 810+705x+95x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 810\mathrm{d}x+\int 705x\mathrm{d}x+\int 95x^{2}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int 810\mathrm{d}x+705\int x\mathrm{d}x+95\int x^{2}\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 810 integralini toping.

$$810x+705\int x\mathrm{d}x+95\int x^{2}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x\mathrm{d}x integralni \frac{x^{2}}{2} bilan almashtiring. 705 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$810x+\frac{705x^{2}}{2}+95\int x^{2}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{2}\mathrm{d}x integralni \frac{x^{3}}{3} bilan almashtiring. 95 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$810x+\frac{705x^{2}}{2}+\frac{95x^{3}}{3}$$

Xos integral bu integral hisoblashning yuqori chegarasida hisoblangan ifodaning boshlangʻich holatidan chiqarib tashlagan holda integral hisoblashning quyi chegarasida hisoblangan ifodaning boshlangʻich holatidir.

$$810\times 3+\frac{705}{2}\times 3^{2}+\frac{95}{3}\times 3^{3}-\left(810\times 0+\frac{705}{2}\times 0^{2}+\frac{95}{3}\times 0^{3}\right)$$

Qisqartirish.

$$\frac{12915}{2}$$