∫ (10x^5-x^3-14x^2-1/2x+3) wrt x eching

$$\int{ (10 { x }^{ 5 } - { x }^{ 3 } -14 { x }^{ 2 } - \frac{ 1 }{ 2 } x+3) }d x$$

Baholash

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{x^{2}}{4}+3x+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int 10x^{5}\mathrm{d}x+\int -x^{3}\mathrm{d}x+\int -14x^{2}\mathrm{d}x+\int -\frac{x}{2}\mathrm{d}x+\int 3\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$10\int x^{5}\mathrm{d}x-\int x^{3}\mathrm{d}x-14\int x^{2}\mathrm{d}x-\frac{\int x\mathrm{d}x}{2}+\int 3\mathrm{d}x$$

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. 10 ni \frac{x^{6}}{6} marotabaga ko'paytirish.

$$\frac{5x^{6}}{3}-\int x^{3}\mathrm{d}x-14\int x^{2}\mathrm{d}x-\frac{\int x\mathrm{d}x}{2}+\int 3\mathrm{d}x$$

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. -1 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-14\int x^{2}\mathrm{d}x-\frac{\int x\mathrm{d}x}{2}+\int 3\mathrm{d}x$$

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. -14 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{\int x\mathrm{d}x}{2}+\int 3\mathrm{d}x$$

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

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{x^{2}}{4}+\int 3\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 3 integralini toping.

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{x^{2}}{4}+3x$$

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.

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{x^{2}}{4}+3x+С$$

x ga nisbatan hosilani topish

$$10x^{5}-x^{3}-14x^{2}-\frac{x}{2}+3$$