∫ 20x(2x+3)^3 wrt x eching

$$\int 20 x ( 2 x + 3 ) ^ { 3 } d x$$

Baholash

$$32x^{5}+180x^{4}+360x^{3}+270x^{2}+С$$

Yechim qadamlari

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

$$\int 20x\left(8x^{3}+36x^{2}+54x+27\right)\mathrm{d}x$$

20x ga 8x^{3}+36x^{2}+54x+27 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int 160x^{4}+720x^{3}+1080x^{2}+540x\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 160x^{4}\mathrm{d}x+\int 720x^{3}\mathrm{d}x+\int 1080x^{2}\mathrm{d}x+\int 540x\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$160\int x^{4}\mathrm{d}x+720\int x^{3}\mathrm{d}x+1080\int x^{2}\mathrm{d}x+540\int x\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{4}\mathrm{d}x integralni \frac{x^{5}}{5} bilan almashtiring. 160 ni \frac{x^{5}}{5} marotabaga ko'paytirish.

$$32x^{5}+720\int x^{3}\mathrm{d}x+1080\int x^{2}\mathrm{d}x+540\int x\mathrm{d}x$$

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

$$32x^{5}+180x^{4}+1080\int x^{2}\mathrm{d}x+540\int x\mathrm{d}x$$

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. 1080 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$32x^{5}+180x^{4}+360x^{3}+540\int x\mathrm{d}x$$

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

$$32x^{5}+180x^{4}+360x^{3}+270x^{2}$$

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.

$$270x^{2}+360x^{3}+180x^{4}+32x^{5}+С$$

x ga nisbatan hosilani topish

$$20x\left(2x+3\right)^{3}$$