∫ (2x^2+3)^3 wrt x eching

$$\int ( 2 x ^ { 2 } + 3 ) ^ { 3 } d x$$

Baholash

$$\frac{8x^{7}}{7}+\frac{36x^{5}}{5}+18x^{3}+27x+С$$

Yechim qadamlari

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

$$\int 8\left(x^{2}\right)^{3}+36\left(x^{2}\right)^{2}+54x^{2}+27\mathrm{d}x$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 3 ni ko‘paytirib, 6 ni oling.

$$\int 8x^{6}+36\left(x^{2}\right)^{2}+54x^{2}+27\mathrm{d}x$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$\int 8x^{6}+36x^{4}+54x^{2}+27\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 8x^{6}\mathrm{d}x+\int 36x^{4}\mathrm{d}x+\int 54x^{2}\mathrm{d}x+\int 27\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$8\int x^{6}\mathrm{d}x+36\int x^{4}\mathrm{d}x+54\int x^{2}\mathrm{d}x+\int 27\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{6}\mathrm{d}x integralni \frac{x^{7}}{7} bilan almashtiring. 8 ni \frac{x^{7}}{7} marotabaga ko'paytirish.

$$\frac{8x^{7}}{7}+36\int x^{4}\mathrm{d}x+54\int x^{2}\mathrm{d}x+\int 27\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{4}\mathrm{d}x integralni \frac{x^{5}}{5} bilan almashtiring. 36 ni \frac{x^{5}}{5} marotabaga ko'paytirish.

$$\frac{8x^{7}}{7}+\frac{36x^{5}}{5}+54\int x^{2}\mathrm{d}x+\int 27\mathrm{d}x$$

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

$$\frac{8x^{7}}{7}+\frac{36x^{5}}{5}+18x^{3}+\int 27\mathrm{d}x$$

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

$$\frac{8x^{7}}{7}+\frac{36x^{5}}{5}+18x^{3}+27x$$

Qisqartirish.

$$27x+18x^{3}+\frac{36x^{5}}{5}+\frac{8x^{7}}{7}$$

Агар F\left(x\right)f\left(x\right) ning dastlabki holati boʻlsa, u holatda f\left(x\right) ning barcha dastlabki holatlari toʻplami F\left(x\right)+C tarafidan belgilanadi. Shu sababli natijaga C\in \mathrm{R} integrallash konstantasini qoʻshing.

$$27x+18x^{3}+\frac{36x^{5}}{5}+\frac{8x^{7}}{7}+С$$

x ga nisbatan hosilani topish

$$\left(2x^{2}+3\right)^{3}$$