∫ (12e^t+7t)dt eching

$$\int ( 12 e ^ { t } + 7 t ) d t$$

Baholash

$$12e^{t}+\frac{7t^{2}}{2}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int 12e^{t}\mathrm{d}t+\int 7t\mathrm{d}t$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$12\int e^{t}\mathrm{d}t+7\int t\mathrm{d}t$$

Natijani olish uchun umumiy integrallar jadvalidagi \int e^{t}\mathrm{d}t=e^{t} integralidan foydalaning.

$$12e^{t}+7\int t\mathrm{d}t$$

k\neq -1 uchun integral \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} boʻlgani uchun, \int t\mathrm{d}t integralni \frac{t^{2}}{2} bilan almashtiring. 7 ni \frac{t^{2}}{2} marotabaga ko'paytirish.

$$12e^{t}+\frac{7t^{2}}{2}$$

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.

$$12e^{t}+\frac{7t^{2}}{2}+С$$

t ga nisbatan hosilani topish

$$7t+12e^{t}$$