∫ left(4x-2right)^3 wrt x eching

$$\int{ { \left(4x-2 \right) }^{ 3 } }d x$$

Baholash

$$\left(2x-1\right)^{4}+С$$

Yechim qadamlari

\left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} binom teoremasini \left(4x-2\right)^{3} kengaytirilishi uchun ishlating.

$$\int 64x^{3}-96x^{2}+48x-8\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 64x^{3}\mathrm{d}x+\int -96x^{2}\mathrm{d}x+\int 48x\mathrm{d}x+\int -8\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$64\int x^{3}\mathrm{d}x-96\int x^{2}\mathrm{d}x+48\int x\mathrm{d}x+\int -8\mathrm{d}x$$

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

$$16x^{4}-96\int x^{2}\mathrm{d}x+48\int x\mathrm{d}x+\int -8\mathrm{d}x$$

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. -96 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$16x^{4}-32x^{3}+48\int x\mathrm{d}x+\int -8\mathrm{d}x$$

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

$$16x^{4}-32x^{3}+24x^{2}+\int -8\mathrm{d}x$$

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

$$16x^{4}-32x^{3}+24x^{2}-8x$$

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.

$$16x^{4}-32x^{3}+24x^{2}-8x+С$$

x ga nisbatan hosilani topish

$$8\left(2x-1\right)^{3}$$