∫ left(3x+2right)^3 wrt x eching

$$\int{ { \left(3x+2 \right) }^{ 3 } }d x$$

Baholash

$$\frac{\left(3x+2\right)^{4}}{12}+С$$

Yechim qadamlari

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

$$\int 27x^{3}+54x^{2}+36x+8\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 27x^{3}\mathrm{d}x+\int 54x^{2}\mathrm{d}x+\int 36x\mathrm{d}x+\int 8\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$27\int x^{3}\mathrm{d}x+54\int x^{2}\mathrm{d}x+36\int x\mathrm{d}x+\int 8\mathrm{d}x$$

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

$$\frac{27x^{4}}{4}+54\int x^{2}\mathrm{d}x+36\int x\mathrm{d}x+\int 8\mathrm{d}x$$

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

$$\frac{27x^{4}}{4}+18x^{3}+36\int x\mathrm{d}x+\int 8\mathrm{d}x$$

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

$$\frac{27x^{4}}{4}+18x^{3}+18x^{2}+\int 8\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 8 integralini toping.

$$\frac{27x^{4}}{4}+18x^{3}+18x^{2}+8x$$

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.

$$\frac{27x^{4}}{4}+18x^{3}+18x^{2}+8x+С$$

x ga nisbatan hosilani topish

$$\left(3x+2\right)^{3}$$