∫ (2sqrt{x}-sqrt[4]{x}) wrt x eching

$$\int ( 2 \sqrt { x } - \sqrt[ 4 ] { x } ) d x$$

Baholash

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

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int 2\sqrt{x}\mathrm{d}x+\int -\sqrt[4]{x}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$2\int \sqrt{x}\mathrm{d}x-\int \sqrt[4]{x}\mathrm{d}x$$

\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.

$$\frac{4x^{\frac{3}{2}}}{3}-\int \sqrt[4]{x}\mathrm{d}x$$

\sqrt[4]{x} ni x^{\frac{1}{4}} 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}{4}}\mathrm{d}x integralni \frac{x^{\frac{5}{4}}}{\frac{5}{4}} bilan almashtiring. Qisqartirish. -1 ni \frac{4x^{\frac{5}{4}}}{5} marotabaga ko'paytirish.

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

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.

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

x ga nisbatan hosilani topish

$$2\sqrt{x}-\sqrt[4]{x}$$