∫ (3x^5-3x^3+1) wrt x eching

$$\int{ (3 { x }^{ 5 } -3 { x }^{ 3 } +1) }d x$$

Baholash

$$\frac{x^{6}}{2}-\frac{3x^{4}}{4}+x+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int 3x^{5}\mathrm{d}x+\int -3x^{3}\mathrm{d}x+\int 1\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$3\int x^{5}\mathrm{d}x-3\int x^{3}\mathrm{d}x+\int 1\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{5}\mathrm{d}x integralni \frac{x^{6}}{6} bilan almashtiring. 3 ni \frac{x^{6}}{6} marotabaga ko'paytirish.

$$\frac{x^{6}}{2}-3\int x^{3}\mathrm{d}x+\int 1\mathrm{d}x$$

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.

$$\frac{x^{6}}{2}-\frac{3x^{4}}{4}+\int 1\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 1 integralini toping.

$$\frac{x^{6}}{2}-\frac{3x^{4}}{4}+x$$

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.

$$\frac{x^{6}}{2}-\frac{3x^{4}}{4}+x+С$$

x ga nisbatan hosilani topish

$$3x^{5}-3x^{3}+1$$