∫ (from - 2 to 5) of (4x^2-2x+1) wrt x eching

$$\int _ { - 2 } ^ { 5 } ( 4 x ^ { 2 } - 2 x + 1 ) d x$$

Baholash

$$\frac{490}{3}\approx 163,333333333$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int 4x^{2}-2x+1\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 4x^{2}\mathrm{d}x+\int -2x\mathrm{d}x+\int 1\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$4\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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

$$\frac{4x^{3}}{3}-2\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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.

$$\frac{4x^{3}}{3}-x^{2}+\int 1\mathrm{d}x$$

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

$$\frac{4x^{3}}{3}-x^{2}+x$$

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

$$\frac{4}{3}\times 5^{3}-5^{2}+5-\left(\frac{4}{3}\left(-2\right)^{3}-\left(-2\right)^{2}-2\right)$$

Qisqartirish.

$$\frac{490}{3}$$