∫ _{frac{sqrt{2}}{2}}^1(x^4-1/2x^4) wrt x eching

$$\int _ { \frac { \sqrt { 2 } } { 2 } } ^ { 1 } ( x ^ { 4 } - \frac { 1 } { 2 } x ^ { 4 } ) d x$$

Baholash

$$-\frac{\sqrt{2}}{80}+\frac{1}{10}\approx 0,08232233$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int x^{4}-\frac{x^{4}}{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int x^{4}\mathrm{d}x+\int -\frac{x^{4}}{2}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x^{4}\mathrm{d}x-\frac{\int x^{4}\mathrm{d}x}{2}$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{4}\mathrm{d}x integralni \frac{x^{5}}{5} bilan almashtiring.

$$\frac{x^{5}}{5}-\frac{\int x^{4}\mathrm{d}x}{2}$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{4}\mathrm{d}x integralni \frac{x^{5}}{5} bilan almashtiring. -\frac{1}{2} ni \frac{x^{5}}{5} marotabaga ko'paytirish.

$$\frac{x^{5}}{5}-\frac{x^{5}}{10}$$

Qisqartirish.

$$\frac{x^{5}}{10}$$

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

$$\frac{1^{5}}{10}-\frac{1}{10}\times \left(\frac{1}{2}\times 2^{\frac{1}{2}}\right)^{5}$$

Qisqartirish.

$$\frac{1}{10}-\frac{\sqrt{2}}{80}$$