∫ (from 0 to 100) of (x^2-7x+9) wrt x eching

$$\int _ { 0 } ^ { 100 } ( x ^ { 2 } - 7 x + 9 ) d x$$

Baholash

$$\frac{897700}{3}\approx 299233,333333333$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int x^{2}-7x+9\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int x^{2}\mathrm{d}x+\int -7x\mathrm{d}x+\int 9\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x^{2}\mathrm{d}x-7\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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.

$$\frac{x^{3}}{3}-7\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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. -7 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{x^{3}}{3}-\frac{7x^{2}}{2}+\int 9\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 9 integralini toping.

$$\frac{x^{3}}{3}-\frac{7x^{2}}{2}+9x$$

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

$$\frac{100^{3}}{3}-\frac{7}{2}\times 100^{2}+9\times 100-\left(\frac{0^{3}}{3}-\frac{7}{2}\times 0^{2}+9\times 0\right)$$

Qisqartirish.

$$\frac{897700}{3}$$