∫ (from 1 to 2) of (x^3+5)^2(3x^2) wrt x eching

$$\int _ { 1 } ^ { 2 } ( x ^ { 3 } + 5 ) ^ { 2 } ( 3 x ^ { 2 } ) d x$$

Baholash

$$\frac{1981}{3}\approx 660,333333333$$

Yechim qadamlari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x^{3}+5\right)^{2} kengaytirilishi uchun ishlating.

$$\int _{1}^{2}\left(\left(x^{3}\right)^{2}+10x^{3}+25\right)\times 3x^{2}\mathrm{d}x$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 3 va 2 ni ko‘paytirib, 6 ni oling.

$$\int _{1}^{2}\left(x^{6}+10x^{3}+25\right)\times 3x^{2}\mathrm{d}x$$

x^{6}+10x^{3}+25 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int _{1}^{2}\left(3x^{6}+30x^{3}+75\right)x^{2}\mathrm{d}x$$

3x^{6}+30x^{3}+75 ga x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int _{1}^{2}3x^{8}+30x^{5}+75x^{2}\mathrm{d}x$$

Avval noaniq integralni baholang.

$$\int 3x^{8}+30x^{5}+75x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 3x^{8}\mathrm{d}x+\int 30x^{5}\mathrm{d}x+\int 75x^{2}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$3\int x^{8}\mathrm{d}x+30\int x^{5}\mathrm{d}x+75\int x^{2}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{8}\mathrm{d}x integralni \frac{x^{9}}{9} bilan almashtiring. 3 ni \frac{x^{9}}{9} marotabaga ko'paytirish.

$$\frac{x^{9}}{3}+30\int x^{5}\mathrm{d}x+75\int x^{2}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{5}\mathrm{d}x integralni \frac{x^{6}}{6} bilan almashtiring. 30 ni \frac{x^{6}}{6} marotabaga ko'paytirish.

$$\frac{x^{9}}{3}+5x^{6}+75\int x^{2}\mathrm{d}x$$

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. 75 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$\frac{x^{9}}{3}+5x^{6}+25x^{3}$$

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

$$25\times 2^{3}+5\times 2^{6}+\frac{2^{9}}{3}-\left(25\times 1^{3}+5\times 1^{6}+\frac{1^{9}}{3}\right)$$

Qisqartirish.

$$\frac{1981}{3}$$