∫ (2x-5)(3x+1) wrt x eching

$$\int ( 2 x - 5 ) ( 3 x + 1 ) d x$$

Baholash

$$2x^{3}-\frac{13x^{2}}{2}-5x+С$$

Yechim qadamlari

2x-5 ifodaning har bir elementini 3x+1 ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$\int 6x^{2}+2x-15x-5\mathrm{d}x$$

-13x ni olish uchun 2x va -15x ni birlashtirish.

$$\int 6x^{2}-13x-5\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 6x^{2}\mathrm{d}x+\int -13x\mathrm{d}x+\int -5\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$6\int x^{2}\mathrm{d}x-13\int x\mathrm{d}x+\int -5\mathrm{d}x$$

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.

$$2x^{3}-13\int x\mathrm{d}x+\int -5\mathrm{d}x$$

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. -13 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$2x^{3}-\frac{13x^{2}}{2}+\int -5\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, -5 integralini toping.

$$2x^{3}-\frac{13x^{2}}{2}-5x$$

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.

$$2x^{3}-\frac{13x^{2}}{2}-5x+С$$

x ga nisbatan hosilani topish

$$\left(2x-5\right)\left(3x+1\right)$$