∫ (2x^3-4x+1) wrt x eching

$$\int ( 2 x ^ { 3 } - 4 x + 1 ) d x$$

Baholash

$$\frac{x^{4}}{2}-2x^{2}+x+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int 2x^{3}\mathrm{d}x+\int -4x\mathrm{d}x+\int 1\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$2\int x^{3}\mathrm{d}x-4\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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

$$\frac{x^{4}}{2}-4\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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. -4 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{x^{4}}{2}-2x^{2}+\int 1\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 1 integralini toping.

$$\frac{x^{4}}{2}-2x^{2}+x$$

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.

$$\frac{x^{4}}{2}-2x^{2}+x+С$$

x ga nisbatan hosilani topish

$$2x^{3}-4x+1$$