∫ (frac{sqrt{x}}{5}+4/sqrt[4]{x}) wrt x eching

$$\int ( \frac { \sqrt { x } } { 5 } + \frac { 4 } { \sqrt[ 4 ] { x } } ) d x$$

Baholash

$$\frac{2x^{\frac{3}{2}}}{15}+\frac{16x^{\frac{3}{4}}}{3}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int \frac{\sqrt{x}}{5}\mathrm{d}x+\int \frac{4}{\sqrt[4]{x}}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\frac{\int \sqrt{x}\mathrm{d}x}{5}+4\int \frac{1}{\sqrt[4]{x}}\mathrm{d}x$$

\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. \frac{1}{5} ni \frac{2x^{\frac{3}{2}}}{3} marotabaga ko'paytirish.

$$\frac{2x^{\frac{3}{2}}}{15}+4\int \frac{1}{\sqrt[4]{x}}\mathrm{d}x$$

\frac{1}{\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{3}{4}}}{\frac{3}{4}} bilan almashtiring. Qisqartirish. 4 ni \frac{4x^{\frac{3}{4}}}{3} marotabaga ko'paytirish.

$$\frac{2x^{\frac{3}{2}}}{15}+\frac{16x^{\frac{3}{4}}}{3}$$

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.

$$\frac{2x^{\frac{3}{2}}}{15}+\frac{16x^{\frac{3}{4}}}{3}+С$$

x ga nisbatan hosilani topish

$$\frac{\sqrt{x}}{5}+\frac{4}{\sqrt[4]{x}}$$