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

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

Baholash

$$\frac{62}{3}\approx 20,666666667$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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.

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

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.

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

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

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

Qisqartirish.

$$\frac{62}{3}$$