∫ (3v^5-v^3/3)dv eching

$$\int ( 3 v ^ { 5 } - v ^ { 3 / 3 } ) d v$$

Baholash

$$\frac{v^{6}}{2}-\frac{v^{2}}{2}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int 3v^{5}\mathrm{d}v+\int -v\mathrm{d}v$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$3\int v^{5}\mathrm{d}v-\int v\mathrm{d}v$$

k\neq -1 uchun integral \int v^{k}\mathrm{d}v=\frac{v^{k+1}}{k+1} boʻlgani uchun, \int v^{5}\mathrm{d}v integralni \frac{v^{6}}{6} bilan almashtiring. 3 ni \frac{v^{6}}{6} marotabaga ko'paytirish.

$$\frac{v^{6}}{2}-\int v\mathrm{d}v$$

k\neq -1 uchun integral \int v^{k}\mathrm{d}v=\frac{v^{k+1}}{k+1} boʻlgani uchun, \int v\mathrm{d}v integralni \frac{v^{2}}{2} bilan almashtiring. -1 ni \frac{v^{2}}{2} marotabaga ko'paytirish.

$$\frac{v^{6}-v^{2}}{2}$$

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

$$\frac{v^{6}}{2}-\frac{v^{2}}{2}+С$$

v ga nisbatan hosilani topish

$$v\left(3v^{4}-1\right)$$