∫ x^3(2x^-2-3) wrt x eching

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

Baholash

$$-\frac{3x^{4}}{4}+x^{2}+С$$

Yechim qadamlari

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

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

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 3 va -2 ni qo‘shib, 1 ni oling.

$$\int 2x^{1}-3x^{3}\mathrm{d}x$$

1 daraja ko‘rsatkichini x ga hisoblang va x ni qiymatni oling.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x\mathrm{d}x integralni \frac{x^{2}}{2} bilan almashtiring. 2 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

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

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. -3 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

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

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.

$$x^{2}-\frac{3x^{4}}{4}+С$$

x ga nisbatan hosilani topish

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