∫ (x+1)(2x+2)(x+1) eching

$$\int ( x + 1 ) ( 2 x + 2 ) ( x + 1 )$$

Baholash

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

Yechim qadamlari

\left(x+1\right)^{2} hosil qilish uchun x+1 va x+1 ni ko'paytirish.

$$\int \left(x+1\right)^{2}\left(2x+2\right)\mathrm{d}x$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$\int \left(x^{2}+2x+1\right)\left(2x+2\right)\mathrm{d}x$$

x^{2}+2x+1 ifodaning har bir elementini 2x+2 ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

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

6x^{2} ni olish uchun 2x^{2} va 4x^{2} ni birlashtirish.

$$\int 2x^{3}+6x^{2}+4x+2x+2\mathrm{d}x$$

6x ni olish uchun 4x va 2x ni birlashtirish.

$$\int 2x^{3}+6x^{2}+6x+2\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 2x^{3}\mathrm{d}x+\int 6x^{2}\mathrm{d}x+\int 6x\mathrm{d}x+\int 2\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$2\int x^{3}\mathrm{d}x+6\int x^{2}\mathrm{d}x+6\int x\mathrm{d}x+\int 2\mathrm{d}x$$

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

$$\frac{x^{4}}{2}+6\int x^{2}\mathrm{d}x+6\int x\mathrm{d}x+\int 2\mathrm{d}x$$

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

$$\frac{x^{4}}{2}+2x^{3}+6\int x\mathrm{d}x+\int 2\mathrm{d}x$$

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

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

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 2 integralini toping.

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

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.

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

x ga nisbatan hosilani topish

$$2\left(x+1\right)^{3}$$