∫ (from - 2 to 5) of (4x-3)^2 wrt x eching

$$\int _ { - 2 } ^ { 5 } ( 4 x - 3 ) ^ { 2 } d x$$

Baholash

$$\frac{1561}{3}\approx 520,333333333$$

Yechim qadamlari

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

$$\int _{-2}^{5}16x^{2}-24x+9\mathrm{d}x$$

Avval noaniq integralni baholang.

$$\int 16x^{2}-24x+9\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 16x^{2}\mathrm{d}x+\int -24x\mathrm{d}x+\int 9\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$16\int x^{2}\mathrm{d}x-24\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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.

$$\frac{16x^{3}}{3}-24\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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. -24 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{16x^{3}}{3}-12x^{2}+\int 9\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 9 integralini toping.

$$\frac{16x^{3}}{3}-12x^{2}+9x$$

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

$$\frac{16}{3}\times 5^{3}-12\times 5^{2}+9\times 5-\left(\frac{16}{3}\left(-2\right)^{3}-12\left(-2\right)^{2}+9\left(-2\right)\right)$$

Qisqartirish.

$$\frac{1561}{3}$$