∫ (from 1 to 15) of 2x^2-5x+3 wrt x eching

$$\int _ { 1 } ^ { 15 } 2 x ^ { 2 } - 5 x + 3 d x$$

Baholash

$$\frac{5194}{3}\approx 1731,333333333$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int 2x^{2}-5x+3\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 2x^{2}\mathrm{d}x+\int -5x\mathrm{d}x+\int 3\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$2\int x^{2}\mathrm{d}x-5\int x\mathrm{d}x+\int 3\mathrm{d}x$$

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.

$$\frac{2x^{3}}{3}-5\int x\mathrm{d}x+\int 3\mathrm{d}x$$

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. -5 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{2x^{3}}{3}-\frac{5x^{2}}{2}+\int 3\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 3 integralini toping.

$$\frac{2x^{3}}{3}-\frac{5x^{2}}{2}+3x$$

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

$$\frac{2}{3}\times 15^{3}-\frac{5}{2}\times 15^{2}+3\times 15-\left(\frac{2}{3}\times 1^{3}-\frac{5}{2}\times 1^{2}+3\times 1\right)$$

Qisqartirish.

$$\frac{5194}{3}$$