∫ (from 4 to 8) of (x^2-12x+32) wrt x eching

$$\int _ { 4 } ^ { 8 } ( x ^ { 2 } - 12 x + 32 ) d x$$

Baholash

$$-\frac{32}{3}\approx -10,666666667$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int x^{2}-12x+32\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int x^{2}\mathrm{d}x+\int -12x\mathrm{d}x+\int 32\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x^{2}\mathrm{d}x-12\int x\mathrm{d}x+\int 32\mathrm{d}x$$

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.

$$\frac{x^{3}}{3}-12\int x\mathrm{d}x+\int 32\mathrm{d}x$$

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. -12 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{x^{3}}{3}-6x^{2}+\int 32\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 32 integralini toping.

$$\frac{x^{3}}{3}-6x^{2}+32x$$

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

$$\frac{8^{3}}{3}-6\times 8^{2}+32\times 8-\left(\frac{4^{3}}{3}-6\times 4^{2}+32\times 4\right)$$

Qisqartirish.

$$-\frac{32}{3}$$