∫ (9/sqrt[4]{t}+4/t^7)dt eching

$$\int ( \frac { 9 } { \sqrt[ 4 ] { t } } + \frac { 4 } { t ^ { 7 } } ) d t$$

Baholash

$$12t^{\frac{3}{4}}-\frac{2}{3t^{6}}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int \frac{9}{\sqrt[4]{t}}\mathrm{d}t+\int \frac{4}{t^{7}}\mathrm{d}t$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$9\int \frac{1}{\sqrt[4]{t}}\mathrm{d}t+4\int \frac{1}{t^{7}}\mathrm{d}t$$

\frac{1}{\sqrt[4]{t}} ni t^{-\frac{1}{4}} 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}{4}}\mathrm{d}t integralni \frac{t^{\frac{3}{4}}}{\frac{3}{4}} bilan almashtiring. Qisqartirish. 9 ni \frac{4t^{\frac{3}{4}}}{3} marotabaga ko'paytirish.

$$12t^{\frac{3}{4}}+4\int \frac{1}{t^{7}}\mathrm{d}t$$

k\neq -1 uchun integral \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} boʻlgani uchun, \int \frac{1}{t^{7}}\mathrm{d}t integralni -\frac{1}{6t^{6}} bilan almashtiring. 4 ni -\frac{1}{6t^{6}} marotabaga ko'paytirish.

$$12t^{\frac{3}{4}}-\frac{2}{3t^{6}}$$

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.

$$12t^{\frac{3}{4}}-\frac{2}{3t^{6}}+С$$

t ga nisbatan hosilani topish

$$\frac{9}{\sqrt[4]{t}}+\frac{4}{t^{7}}$$