∫ (from 0 to 1) of (3x^3-x^2+2x-4) wrt x eching

$$\int_{ 0 }^{ 1 } (3 { x }^{ 3 } - { x }^{ 2 } +2x-4) d x$$

Baholash

$$-\frac{31}{12}\approx -2,583333333$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{3}\mathrm{d}x integralni \frac{x^{4}}{4} bilan almashtiring. 3 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

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

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. -1 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

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

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

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

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, -4 integralini toping.

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

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

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

Qisqartirish.

$$-\frac{31}{12}$$