∫ x+2x^2 wrt x eching

$$\int{ x+2 { x }^{ 2 } }d x$$

Baholash

$$\frac{2x^{3}}{3}+\frac{x^{2}}{2}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int x\mathrm{d}x+\int 2x^{2}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x\mathrm{d}x+2\int x^{2}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x\mathrm{d}x integralni \frac{x^{2}}{2} bilan almashtiring.

$$\frac{x^{2}}{2}+2\int x^{2}\mathrm{d}x$$

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

$$\frac{x^{2}}{2}+\frac{2x^{3}}{3}$$

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.

$$\frac{x^{2}}{2}+\frac{2x^{3}}{3}+С$$

x ga nisbatan hosilani topish

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