∫ x-2sqrt{x}+1 wrt x eching

$$\int{ x-2 \sqrt{ x } +1 }d x$$

Baholash

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

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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.

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

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

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

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

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

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.

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

x ga nisbatan hosilani topish

$$\left(\sqrt{x}-1\right)^{2}$$