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

$$\int_{ 1 }^{ 3 } (3 { x }^{ 3 } -2x+5) d x$$

Baholash

$$62$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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

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

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.

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

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 5 integralini toping.

$$\frac{3x^{4}}{4}-x^{2}+5x$$

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

$$\frac{3}{4}\times 3^{4}-3^{2}+5\times 3-\left(\frac{3}{4}\times 1^{4}-1^{2}+5\times 1\right)$$

Qisqartirish.

$$62$$