∫ (-7sqrt{x}+5sqrt[4]{x}) wrt x eching

$$\int ( - 7 \sqrt { x } + 5 \sqrt[ 4 ] { x } ) d x$$

Baholash

$$-\frac{14x^{\frac{3}{2}}}{3}+4x^{\frac{5}{4}}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int -7\sqrt{x}\mathrm{d}x+\int 5\sqrt[4]{x}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$-7\int \sqrt{x}\mathrm{d}x+5\int \sqrt[4]{x}\mathrm{d}x$$

\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. -7 ni \frac{2x^{\frac{3}{2}}}{3} marotabaga ko'paytirish.

$$-\frac{14x^{\frac{3}{2}}}{3}+5\int \sqrt[4]{x}\mathrm{d}x$$

\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. 5 ni \frac{4x^{\frac{5}{4}}}{5} marotabaga ko'paytirish.

$$-\frac{14x^{\frac{3}{2}}}{3}+4x^{\frac{5}{4}}$$

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.

$$-\frac{14x^{\frac{3}{2}}}{3}+4x^{\frac{5}{4}}+С$$

x ga nisbatan hosilani topish

$$-7\sqrt{x}+5\sqrt[4]{x}$$