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

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

Baholash

$$\frac{1}{3}\approx 0,333333333$$

Yechim qadamlari

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

$$\int _{2}^{3}x^{2}-4x+4\mathrm{d}x$$

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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.

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

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. -4 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{x^{3}}{3}-2x^{2}+\int 4\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 4 integralini toping.

$$\frac{x^{3}}{3}-2x^{2}+4x$$

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

$$\frac{3^{3}}{3}-2\times 3^{2}+4\times 3-\left(\frac{2^{3}}{3}-2\times 2^{2}+4\times 2\right)$$

Qisqartirish.

$$\frac{1}{3}$$