∫ (from - 1 to 1) of (1-y)y wrt y eching

$$\int _ { - 1 } ^ { 1 } ( 1 - y ) y d y$$

Baholash

$$-\frac{2}{3}\approx -0,666666667$$

Yechim qadamlari

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

$$\int _{-1}^{1}y-y^{2}\mathrm{d}y$$

Avval noaniq integralni baholang.

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

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.

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

Xos integral bu integral hisoblashning yuqori chegarasida hisoblangan ifodaning boshlangʻich holatidan chiqarib tashlagan holda integral hisoblashning quyi chegarasida hisoblangan ifodaning boshlangʻich holatidir.

$$\frac{1^{2}}{2}-\frac{1^{3}}{3}-\left(\frac{\left(-1\right)^{2}}{2}-\frac{\left(-1\right)^{3}}{3}\right)$$

Qisqartirish.

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