∫ (x^3-3x+5) wrt x eching

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

Baholash

$$\frac{x^{4}}{4}-\frac{3x^{2}}{2}+5x+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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.

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

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. -3 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

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

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 5 integralini toping.

$$\frac{x^{4}}{4}-\frac{3x^{2}}{2}+5x$$

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.

$$\frac{x^{4}}{4}-\frac{3x^{2}}{2}+5x+С$$

x ga nisbatan hosilani topish

$$x^{3}-3x+5$$