∫ (from 0 to 1) of 1/3-1/3y^3 wrt y eching

$$\int _ { 0 } ^ { 1 } \frac { 1 } { 3 } - \frac { 1 } { 3 } y ^ { 3 } d y$$

Baholash

$$\frac{1}{4}=0,25$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int \frac{1-y^{3}}{3}\mathrm{d}y$$

Summani muddatma-muddat integratsiya qiling.

$$\int \frac{1}{3}\mathrm{d}y+\int -\frac{y^{3}}{3}\mathrm{d}y$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int \frac{1}{3}\mathrm{d}y-\frac{\int y^{3}\mathrm{d}y}{3}$$

\int a\mathrm{d}y=ay umumiy integrallar qoidasi jadvalidan foydalanib, \frac{1}{3} integralini toping.

$$\frac{y-\int y^{3}\mathrm{d}y}{3}$$

k\neq -1 uchun integral \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} boʻlgani uchun, \int y^{3}\mathrm{d}y integralni \frac{y^{4}}{4} bilan almashtiring. -\frac{1}{3} ni \frac{y^{4}}{4} marotabaga ko'paytirish.

$$\frac{y}{3}-\frac{y^{4}}{12}$$

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

$$\frac{1}{3}\times 1-\frac{1^{4}}{12}-\left(\frac{1}{3}\times 0-\frac{0^{4}}{12}\right)$$

Qisqartirish.

$$\frac{1}{4}$$