∫ (from 0 to 1) of (x^2-3x+1) wrt x eching

$$\int _ { 0 } ^ { 1 } ( x ^ { 2 } - 3 x + 1 ) d x$$

Baholash

$$-\frac{1}{6}\approx -0,166666667$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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.

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

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. -3 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

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

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

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

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

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

Qisqartirish.

$$-\frac{1}{6}$$