∫ (from 0 to 1) of sqrt{x}-x^2 eching

$$\int _ { 0 } ^ { 1 } \sqrt { x } - x ^ { 2 }$$

Baholash

$$\frac{1}{3}\approx 0,333333333$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int \sqrt{x}-x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int \sqrt{x}\mathrm{d}x-\int x^{2}\mathrm{d}x$$

\sqrt{x} ni x^{\frac{1}{2}} sifatida qaytadan yozish. k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{\frac{1}{2}}\mathrm{d}x integralni \frac{x^{\frac{3}{2}}}{\frac{3}{2}} bilan almashtiring. Qisqartirish.

$$\frac{2x^{\frac{3}{2}}}{3}-\int x^{2}\mathrm{d}x$$

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. -1 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

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

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

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

Qisqartirish.

$$\frac{1}{3}$$