∫ (x-3)^3 wrt x eching

$$\int ( x - 3 ) ^ { 3 } d x$$

Baholash

$$\frac{\left(x-3\right)^{4}}{4}+С$$

Yechim qadamlari

\left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} binom teoremasini \left(x-3\right)^{3} kengaytirilishi uchun ishlating.

$$\int x^{3}-9x^{2}+27x-27\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int x^{3}\mathrm{d}x+\int -9x^{2}\mathrm{d}x+\int 27x\mathrm{d}x+\int -27\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x^{3}\mathrm{d}x-9\int x^{2}\mathrm{d}x+27\int x\mathrm{d}x+\int -27\mathrm{d}x$$

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.

$$\frac{x^{4}}{4}-9\int x^{2}\mathrm{d}x+27\int x\mathrm{d}x+\int -27\mathrm{d}x$$

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. -9 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$\frac{x^{4}}{4}-3x^{3}+27\int x\mathrm{d}x+\int -27\mathrm{d}x$$

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

$$\frac{x^{4}}{4}-3x^{3}+\frac{27x^{2}}{2}+\int -27\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, -27 integralini toping.

$$\frac{x^{4}}{4}-3x^{3}+\frac{27x^{2}}{2}-27x$$

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.

$$\frac{x^{4}}{4}-3x^{3}+\frac{27x^{2}}{2}-27x+С$$

x ga nisbatan hosilani topish

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