∫ (from - 2 to 5) of (4x-3)^3 wrt x eching

$$\int _ { - 2 } ^ { 5 } ( 4 x - 3 ) ^ { 3 } d x$$

Baholash

$$4305$$

Yechim qadamlari

\left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} binom teoremasini \left(4x-3\right)^{3} kengaytirilishi uchun ishlating.

$$\int _{-2}^{5}64x^{3}-144x^{2}+108x-27\mathrm{d}x$$

Avval noaniq integralni baholang.

$$\int 64x^{3}-144x^{2}+108x-27\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 64x^{3}\mathrm{d}x+\int -144x^{2}\mathrm{d}x+\int 108x\mathrm{d}x+\int -27\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$64\int x^{3}\mathrm{d}x-144\int x^{2}\mathrm{d}x+108\int x\mathrm{d}x+\int -27\mathrm{d}x$$

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

$$16x^{4}-144\int x^{2}\mathrm{d}x+108\int x\mathrm{d}x+\int -27\mathrm{d}x$$

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. -144 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$16x^{4}-48x^{3}+108\int x\mathrm{d}x+\int -27\mathrm{d}x$$

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

$$16x^{4}-48x^{3}+54x^{2}+\int -27\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, -27 integralini toping.

$$16x^{4}-48x^{3}+54x^{2}-27x$$

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

$$16\times 5^{4}-48\times 5^{3}+54\times 5^{2}-27\times 5-\left(16\left(-2\right)^{4}-48\left(-2\right)^{3}+54\left(-2\right)^{2}-27\left(-2\right)\right)$$

Qisqartirish.

$$4305$$