∫ (from 0 to 2) of (4x-x^3)^2 wrt x eching

$$\int _ { 0 } ^ { 2 } ( 4 x - x ^ { 3 } ) ^ { 2 } d x$$

Baholash

$$\frac{1024}{105}\approx 9,752380952$$

Yechim qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(4x-x^{3}\right)^{2} kengaytirilishi uchun ishlating.

$$\int _{0}^{2}16x^{2}-8xx^{3}+\left(x^{3}\right)^{2}\mathrm{d}x$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 1 va 3 ni qo‘shib, 4 ni oling.

$$\int _{0}^{2}16x^{2}-8x^{4}+\left(x^{3}\right)^{2}\mathrm{d}x$$

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

$$\int _{0}^{2}16x^{2}-8x^{4}+x^{6}\mathrm{d}x$$

Avval noaniq integralni baholang.

$$\int 16x^{2}-8x^{4}+x^{6}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 16x^{2}\mathrm{d}x+\int -8x^{4}\mathrm{d}x+\int x^{6}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$16\int x^{2}\mathrm{d}x-8\int x^{4}\mathrm{d}x+\int x^{6}\mathrm{d}x$$

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

$$\frac{16x^{3}}{3}-8\int x^{4}\mathrm{d}x+\int x^{6}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{4}\mathrm{d}x integralni \frac{x^{5}}{5} bilan almashtiring. -8 ni \frac{x^{5}}{5} marotabaga ko'paytirish.

$$\frac{16x^{3}}{3}-\frac{8x^{5}}{5}+\int x^{6}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{6}\mathrm{d}x integralni \frac{x^{7}}{7} bilan almashtiring.

$$\frac{16x^{3}}{3}-\frac{8x^{5}}{5}+\frac{x^{7}}{7}$$

Qisqartirish.

$$\frac{x^{7}}{7}-\frac{8x^{5}}{5}+\frac{16x^{3}}{3}$$

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

$$\frac{2^{7}}{7}-\frac{8}{5}\times 2^{5}+\frac{16}{3}\times 2^{3}-\left(\frac{0^{7}}{7}-\frac{8}{5}\times 0^{5}+\frac{16}{3}\times 0^{3}\right)$$

Qisqartirish.

$$\frac{1024}{105}$$