∫ (3-x)-(x^2-9) wrt x eching

$$\int{ (3-x)-( { x }^{ 2 } -9) }d x$$

Baholash

$$-\frac{x^{3}}{3}-\frac{x^{2}}{2}+12x+С$$

Yechim qadamlari

x^{2}-9 teskarisini topish uchun har birining teskarisini toping.

$$\int 3-x-x^{2}+9\mathrm{d}x$$

12 olish uchun 3 va 9'ni qo'shing.

$$\int 12-x-x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int 12\mathrm{d}x-\int x\mathrm{d}x-\int x^{2}\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 12 integralini toping.

$$12x-\int x\mathrm{d}x-\int x^{2}\mathrm{d}x$$

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. -1 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$12x-\frac{x^{2}}{2}-\int x^{2}\mathrm{d}x$$

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. -1 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$12x-\frac{x^{2}}{2}-\frac{x^{3}}{3}$$

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.

$$12x-\frac{x^{2}}{2}-\frac{x^{3}}{3}+С$$

x ga nisbatan hosilani topish

$$\left(3-x\right)\left(x+4\right)$$