∫ x^3(2x-5) wrt x eching

$$\int x ^ { 3 } ( 2 x - 5 ) d x$$

Baholash

$$\frac{2x^{5}}{5}-\frac{5x^{4}}{4}+С$$

Yechim qadamlari

x^{3} ga 2x-5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int 2x^{4}-5x^{3}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 2x^{4}\mathrm{d}x+\int -5x^{3}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$2\int x^{4}\mathrm{d}x-5\int x^{3}\mathrm{d}x$$

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. 2 ni \frac{x^{5}}{5} marotabaga ko'paytirish.

$$\frac{2x^{5}}{5}-5\int x^{3}\mathrm{d}x$$

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. -5 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

$$\frac{2x^{5}}{5}-\frac{5x^{4}}{4}$$

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.

$$\frac{2x^{5}}{5}-\frac{5x^{4}}{4}+С$$

x ga nisbatan hosilani topish

$$\left(2x-5\right)x^{3}$$