∫ (from 0 to 1) of 5u^5+3u^2+udu eching

$$\int _ { 0 } ^ { 1 } 5 u ^ { 5 } + 3 u ^ { 2 } + u d u$$

Baholash

$$\frac{7}{3}\approx 2,333333333$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int 5u^{5}+3u^{2}+u\mathrm{d}u$$

Summani muddatma-muddat integratsiya qiling.

$$\int 5u^{5}\mathrm{d}u+\int 3u^{2}\mathrm{d}u+\int u\mathrm{d}u$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$5\int u^{5}\mathrm{d}u+3\int u^{2}\mathrm{d}u+\int u\mathrm{d}u$$

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

$$\frac{5u^{6}}{6}+3\int u^{2}\mathrm{d}u+\int u\mathrm{d}u$$

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

$$\frac{5u^{6}}{6}+u^{3}+\int u\mathrm{d}u$$

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

$$\frac{5u^{6}}{6}+u^{3}+\frac{u^{2}}{2}$$

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

$$\frac{5}{6}\times 1^{6}+1^{3}+\frac{1^{2}}{2}-\left(\frac{5}{6}\times 0^{6}+0^{3}+\frac{0^{2}}{2}\right)$$

Qisqartirish.

$$\frac{7}{3}$$