∫ 12x^2+16*2x+62*3 wrt x eching

$$\int 12 x ^ { 2 } + 16 \cdot 2 x + 62 \cdot 3 d x$$

Baholash

$$4x^{3}+16x^{2}+186x+С$$

Yechim qadamlari

Ko‘paytirishlarni bajaring.

$$\int 12x^{2}+32x+186\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 12x^{2}\mathrm{d}x+\int 32x\mathrm{d}x+\int 186\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$12\int x^{2}\mathrm{d}x+32\int x\mathrm{d}x+\int 186\mathrm{d}x$$

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

$$4x^{3}+32\int x\mathrm{d}x+\int 186\mathrm{d}x$$

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

$$4x^{3}+16x^{2}+\int 186\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 186 integralini toping.

$$4x^{3}+16x^{2}+186x$$

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.

$$4x^{3}+16x^{2}+186x+С$$

x ga nisbatan hosilani topish

$$12x^{2}+32x+186$$