∫ 2x^2(4x-6) wrt x eching

$$\int{ 2 { x }^{ 2 } (4x-6) }d x$$

Baholash

$$2x^{4}-4x^{3}+С$$

Yechim qadamlari

2x^{2} ga 4x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int 8x^{3}-12x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 8x^{3}\mathrm{d}x+\int -12x^{2}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$8\int x^{3}\mathrm{d}x-12\int x^{2}\mathrm{d}x$$

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

$$2x^{4}-12\int x^{2}\mathrm{d}x$$

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. -12 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$2x^{4}-4x^{3}$$

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.

$$2x^{4}-4x^{3}+С$$

x ga nisbatan hosilani topish

$$4\left(2x-3\right)x^{2}$$