∫ (3x^2+x+5) wrt x eching

$$\int ( 3 x ^ { 2 } + x + 5 ) d x$$

Baholash

$$x^{3}+\frac{x^{2}}{2}+5x+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{2}\mathrm{d}x integralni \frac{x^{3}}{3} bilan almashtiring. 3 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

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

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

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

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

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

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.

$$x^{3}+\frac{x^{2}}{2}+5x+С$$

x ga nisbatan hosilani topish

$$3x^{2}+x+5$$