∫ x^3+1/x+1 wrt x eching

$$\int \frac { x ^ { 3 } + 1 } { x + 1 } d x$$

Baholash

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

Yechim qadamlari

\frac{x^{3}+1}{x+1} ichida hali faktorlanmagan ifodalarni faktorlang.

$$\int \frac{\left(x+1\right)\left(x^{2}-x+1\right)}{x+1}\mathrm{d}x$$

Surat va maxrajdagi ikkala x+1 ni qisqartiring.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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

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

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

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

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

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

Qisqartirish.

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

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.

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

x ga nisbatan hosilani topish

$$x^{2}-x+1$$