∫ (x-2)^2(x) wrt x eching

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

Baholash

$$\frac{x^{4}}{4}-\frac{4x^{3}}{3}+2x^{2}+С$$

Yechim qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-2\right)^{2} kengaytirilishi uchun ishlating.

$$\int \left(x^{2}-4x+4\right)x\mathrm{d}x$$

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

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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.

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

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. -4 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

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

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.

$$\frac{x^{4}}{4}-\frac{4x^{3}}{3}+2x^{2}$$

Qisqartirish.

$$2x^{2}-\frac{4x^{3}}{3}+\frac{x^{4}}{4}$$

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.

$$2x^{2}-\frac{4x^{3}}{3}+\frac{x^{4}}{4}+С$$

x ga nisbatan hosilani topish

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