∫ 5x(4-2x^2) eching

$$\int 5 x ( 4 - 2 x ^ { 2 } )$$

Baholash

$$-\frac{5x^{4}}{2}+10x^{2}+С$$

Yechim qadamlari

5x ga 4-2x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int 20x-10x^{3}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 20x\mathrm{d}x+\int -10x^{3}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$20\int x\mathrm{d}x-10\int x^{3}\mathrm{d}x$$

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

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

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. -10 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

$$10x^{2}-\frac{5x^{4}}{2}$$

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.

$$10x^{2}-\frac{5x^{4}}{2}+С$$

x ga nisbatan hosilani topish

$$20x-10x^{3}$$