∫ y(1-y) wrt y eching

$$\int y ( 1 - y ) d y$$

Baholash

$$-\frac{y^{3}}{3}+\frac{y^{2}}{2}+С$$

Yechim qadamlari

y ga 1-y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int y-y^{2}\mathrm{d}y$$

Summani muddatma-muddat integratsiya qiling.

$$\int y\mathrm{d}y+\int -y^{2}\mathrm{d}y$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int y\mathrm{d}y-\int y^{2}\mathrm{d}y$$

k\neq -1 uchun integral \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} boʻlgani uchun, \int y\mathrm{d}y integralni \frac{y^{2}}{2} bilan almashtiring.

$$\frac{y^{2}}{2}-\int y^{2}\mathrm{d}y$$

k\neq -1 uchun integral \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} boʻlgani uchun, \int y^{2}\mathrm{d}y integralni \frac{y^{3}}{3} bilan almashtiring. -1 ni \frac{y^{3}}{3} marotabaga ko'paytirish.

$$\frac{y^{2}}{2}-\frac{y^{3}}{3}$$

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

$$\frac{y^{2}}{2}-\frac{y^{3}}{3}+С$$

y ga nisbatan hosilani topish

$$y\left(1-y\right)$$