∫ (from 0 to 3) of [3x+2x^2-x^3] eching

$$\int _ { 0 } ^ { 3 } [ 3 x + 2 x ^ { 2 } - x ^ { 3 } ]$$

Baholash

$$\frac{45}{4}=11,25$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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.

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

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

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

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. -1 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

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

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

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

Qisqartirish.

$$\frac{45}{4}$$