∫ (from 5 to 9) of x^4-28x^3+286x^2-1260x+2025 wrt x eching

$$\int_{ 5 }^{ 9 } { x }^{ 4 } -28 { x }^{ 3 } +286 { x }^{ 2 } -1260x+2025 d x$$

Baholash

$$\frac{512}{15}\approx 34,133333333$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int x^{4}-28x^{3}+286x^{2}-1260x+2025\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int x^{4}\mathrm{d}x+\int -28x^{3}\mathrm{d}x+\int 286x^{2}\mathrm{d}x+\int -1260x\mathrm{d}x+\int 2025\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x^{4}\mathrm{d}x-28\int x^{3}\mathrm{d}x+286\int x^{2}\mathrm{d}x-1260\int x\mathrm{d}x+\int 2025\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{4}\mathrm{d}x integralni \frac{x^{5}}{5} bilan almashtiring.

$$\frac{x^{5}}{5}-28\int x^{3}\mathrm{d}x+286\int x^{2}\mathrm{d}x-1260\int x\mathrm{d}x+\int 2025\mathrm{d}x$$

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. -28 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

$$\frac{x^{5}}{5}-7x^{4}+286\int x^{2}\mathrm{d}x-1260\int x\mathrm{d}x+\int 2025\mathrm{d}x$$

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

$$\frac{x^{5}}{5}-7x^{4}+\frac{286x^{3}}{3}-1260\int x\mathrm{d}x+\int 2025\mathrm{d}x$$

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. -1260 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{x^{5}}{5}-7x^{4}+\frac{286x^{3}}{3}-630x^{2}+\int 2025\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 2025 integralini toping.

$$\frac{x^{5}}{5}-7x^{4}+\frac{286x^{3}}{3}-630x^{2}+2025x$$

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

$$\frac{9^{5}}{5}-7\times 9^{4}+\frac{286}{3}\times 9^{3}-630\times 9^{2}+2025\times 9-\left(\frac{5^{5}}{5}-7\times 5^{4}+\frac{286}{3}\times 5^{3}-630\times 5^{2}+2025\times 5\right)$$

Qisqartirish.

$$\frac{512}{15}$$