∫ (2x-5)(3x+2) wrt x= eching

$$\int ( 2 x - 5 ) ( 3 x + 2 ) d x =$$

Baholash

$$2x^{3}-\frac{11x^{2}}{2}-10x+С$$

Yechim qadamlari

2x-5 ifodaning har bir elementini 3x+2 ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$\int 6x^{2}+4x-15x-10\mathrm{d}x$$

-11x ni olish uchun 4x va -15x ni birlashtirish.

$$\int 6x^{2}-11x-10\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 6x^{2}\mathrm{d}x+\int -11x\mathrm{d}x+\int -10\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$6\int x^{2}\mathrm{d}x-11\int x\mathrm{d}x+\int -10\mathrm{d}x$$

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.

$$2x^{3}-11\int x\mathrm{d}x+\int -10\mathrm{d}x$$

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. -11 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$2x^{3}-\frac{11x^{2}}{2}+\int -10\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, -10 integralini toping.

$$2x^{3}-\frac{11x^{2}}{2}-10x$$

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.

$$2x^{3}-\frac{11x^{2}}{2}-10x+С$$

x ga nisbatan hosilani topish

$$\left(2x-5\right)\left(3x+2\right)$$