∫ (from 0 to sqrt { 2) of / 2 } (1/2x^2-x^4) wrt x eching

$$\int _ { 0 } ^ { \frac { \sqrt { 2 } } { 2 } } ( \frac { 1 } { 2 } x ^ { 2 } - x ^ { 4 } ) d x$$

Baholash

$$\frac{\sqrt{2}}{60}\approx 0,023570226$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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

$$\frac{x^{3}}{6}-\int x^{4}\mathrm{d}x$$

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. -1 ni \frac{x^{5}}{5} marotabaga ko'paytirish.

$$\frac{x^{3}}{6}-\frac{x^{5}}{5}$$

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

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

Qisqartirish.

$$\frac{\sqrt{2}}{60}$$