∫ (x+1)(2x-1)(x+9) wrt x eching

$$\int ( x + 1 ) ( 2 x - 1 ) ( x + 9 ) d x$$

Baholash

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}-9x+С$$

Yechim qadamlari

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

$$\int \left(2x^{2}-x+2x-1\right)\left(x+9\right)\mathrm{d}x$$

x ni olish uchun -x va 2x ni birlashtirish.

$$\int \left(2x^{2}+x-1\right)\left(x+9\right)\mathrm{d}x$$

2x^{2}+x-1 ifodaning har bir elementini x+9 ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$\int 2x^{3}+18x^{2}+x^{2}+9x-x-9\mathrm{d}x$$

19x^{2} ni olish uchun 18x^{2} va x^{2} ni birlashtirish.

$$\int 2x^{3}+19x^{2}+9x-x-9\mathrm{d}x$$

8x ni olish uchun 9x va -x ni birlashtirish.

$$\int 2x^{3}+19x^{2}+8x-9\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 2x^{3}\mathrm{d}x+\int 19x^{2}\mathrm{d}x+\int 8x\mathrm{d}x+\int -9\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$2\int x^{3}\mathrm{d}x+19\int x^{2}\mathrm{d}x+8\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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.

$$\frac{x^{4}}{2}+19\int x^{2}\mathrm{d}x+8\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+8\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}+\int -9\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, -9 integralini toping.

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}-9x$$

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.

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}-9x+С$$

x ga nisbatan hosilani topish

$$\left(2x-1\right)\left(x+1\right)\left(x+9\right)$$