∫ (from - 1 to 2) of (2x-5)^2 wrt x= eching

$$\int _ { - 1 } ^ { 2 } ( 2 x - 5 ) ^ { 2 } d x =$$

Baholash

$$57$$

Yechim qadamlari

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

$$\int _{-1}^{2}4x^{2}-20x+25\mathrm{d}x$$

Avval noaniq integralni baholang.

$$\int 4x^{2}-20x+25\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 4x^{2}\mathrm{d}x+\int -20x\mathrm{d}x+\int 25\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$4\int x^{2}\mathrm{d}x-20\int x\mathrm{d}x+\int 25\mathrm{d}x$$

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.

$$\frac{4x^{3}}{3}-20\int x\mathrm{d}x+\int 25\mathrm{d}x$$

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. -20 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{4x^{3}}{3}-10x^{2}+\int 25\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 25 integralini toping.

$$\frac{4x^{3}}{3}-10x^{2}+25x$$

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

$$\frac{4}{3}\times 2^{3}-10\times 2^{2}+25\times 2-\left(\frac{4}{3}\left(-1\right)^{3}-10\left(-1\right)^{2}+25\left(-1\right)\right)$$

Qisqartirish.

$$57$$