∫ 3x^3-x^4+2x^5/x^2 wrt x eching

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

Baholash

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

Yechim qadamlari

\frac{3x^{3}-x^{4}+2x^{5}}{x^{2}} ichida hali faktorlanmagan ifodalarni faktorlang.

$$\int \frac{\left(2x^{2}-x+3\right)x^{3}}{x^{2}}\mathrm{d}x$$

Surat va maxrajdagi ikkala x^{2} ni qisqartiring.

$$\int x\left(2x^{2}-x+3\right)\mathrm{d}x$$

Ifodani kengaytiring.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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

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

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. -1 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

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

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.

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

Qisqartirish.

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

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.

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

x ga nisbatan hosilani topish

$$x\left(2x^{2}-x+3\right)$$