∫ (4/sqrt[3]{t}+3/t^6)dt eching

$$\int ( \frac { 4 } { \sqrt[ 3 ] { t } } + \frac { 3 } { t ^ { 6 } } ) d t$$

Baholash

$$6t^{\frac{2}{3}}-\frac{3}{5t^{5}}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int \frac{4}{\sqrt[3]{t}}\mathrm{d}t+\int \frac{3}{t^{6}}\mathrm{d}t$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$4\int \frac{1}{\sqrt[3]{t}}\mathrm{d}t+3\int \frac{1}{t^{6}}\mathrm{d}t$$

\frac{1}{\sqrt[3]{t}} ni t^{-\frac{1}{3}} sifatida qaytadan yozish. k\neq -1 uchun integral \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} boʻlgani uchun, \int t^{-\frac{1}{3}}\mathrm{d}t integralni \frac{t^{\frac{2}{3}}}{\frac{2}{3}} bilan almashtiring. Qisqartirish. 4 ni \frac{3t^{\frac{2}{3}}}{2} marotabaga ko'paytirish.

$$6t^{\frac{2}{3}}+3\int \frac{1}{t^{6}}\mathrm{d}t$$

k\neq -1 uchun integral \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} boʻlgani uchun, \int \frac{1}{t^{6}}\mathrm{d}t integralni -\frac{1}{5t^{5}} bilan almashtiring. 3 ni -\frac{1}{5t^{5}} marotabaga ko'paytirish.

$$6t^{\frac{2}{3}}-\frac{\frac{3}{t^{5}}}{5}$$

Qisqartirish.

$$6t^{\frac{2}{3}}-\frac{3}{5t^{5}}$$

Агар F\left(t\right)f\left(t\right) ning dastlabki holati boʻlsa, u holatda f\left(t\right) ning barcha dastlabki holatlari toʻplami F\left(t\right)+C tarafidan belgilanadi. Shu sababli natijaga C\in \mathrm{R} integrallash konstantasini qoʻshing.

$$6t^{\frac{2}{3}}-\frac{3}{5t^{5}}+С$$

t ga nisbatan hosilani topish

$$\frac{4}{\sqrt[3]{t}}+\frac{3}{t^{6}}$$