∫ (from 0 to 1) of (2x-1)^2 wrt x= eching

$$\int _ { 0 } ^ { 1 } ( 2 x - 1 ) ^ { 2 } d x =$$

Baholash

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

Yechim qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-1\right)^{2} kengaytirilishi uchun ishlating.

$$\int _{0}^{1}4x^{2}-4x+1\mathrm{d}x$$

Avval noaniq integralni baholang.

$$\int 4x^{2}-4x+1\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 4x^{2}\mathrm{d}x+\int -4x\mathrm{d}x+\int 1\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$4\int x^{2}\mathrm{d}x-4\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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

$$\frac{4x^{3}}{3}-4\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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.

$$\frac{4x^{3}}{3}-2x^{2}+\int 1\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 1 integralini toping.

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

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

$$\frac{4}{3}\times 1^{3}-2\times 1^{2}+1-\left(\frac{4}{3}\times 0^{3}-2\times 0^{2}+0\right)$$

Qisqartirish.

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