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

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

Baholash

$$36$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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

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

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

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

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

$$x^{3}+x^{2}+x$$

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

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

Qisqartirish.

$$36$$